January 01, 1970
Let \(p\) be an odd prime, let \(e\ge2\), and put \(q=p^e\). We study the wild family \[\varphi_{r,e}(x)=x^q+qp^r x^{q+1}=x^{p^e}+p^{r+e}x^{p^e+1} \qquad (r\ge0),\] and the inverse Böttcher coordinate \(f_{r,e}(x)=x\sum_{k\ge0}a_k(r,e)x^k/k!\) characterized by \[\varphi_{r,e}(f_{r,e}(x))=f_{r,e}(x^q).\] For the clean family, we prove a complete mod-\(p\) digit-sum law in the special fiber \(r=0\). For the higher fibers \(r\ge1\), we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a lag-\(e\) pure-power recursion, and subadditivity of the induced digit weight. This yields the pure-power branch word \[(B^{e-1}A)^{\lceil r/e\rceil}B^\infty\] and the radius formula \[\rho(f_{r,e})=p^{-\theta_{r,e}},\qquad \theta_{r,e}=p^{-e\lceil r/e\rceil}\left(\frac{1}{p-1}+e\lceil r/e\rceil-r\right).\] We then prove a tail-stable extension. In the special fiber, \(p\)-divisible tails preserve the digit-sum law modulo \(p\). In the higher fibers, tails satisfying \(\mathop{\mathrm{ord}}_p(\vartheta_h)\ge\Lambda_{r,e}(h+1)+1\) lie beyond the clean-family initial \(\Lambda_{r,e}\)-graded term and therefore preserve the leading terms, the pure-power branch word, the valuation asymptotic, and the radius. For \(e=2\), this recovers the Salerno–Silverman degree-\(p^2\) family and the Fu–Nie radius statement for the inverse coordinate in that family.
The local study of superattracting germs begins with Böttcher’s theorem [1], which gives a canonical coordinate in which a germ is conjugate to a pure monomial. In one complex variable this coordinate is central both near a superattracting fixed point and, for polynomials, near infinity, where it encodes the escape-rate function and the external geometry of Julia sets; see, for example, [2]–[6]. The same normal form also appears in higher-dimensional holomorphic dynamics [7], [8].
Over non-archimedean fields, Böttcher coordinates are part of a broader local and global theory of rational dynamics over valued fields; see, for example, [9]–[14]. In this setting the coefficients of the coordinate carry arithmetic information, reflecting ramification, integrality, and valuation phenomena. In polynomial dynamics, \(p\)-adic Böttcher coordinates have also been used in the study of arboreal Galois representations and bounded-height problems in families [15], [16]. Fix an odd prime \(p\), an integer \(e\ge 2\), and put \(q=p^e\). In this paper we study the wild one-parameter family \[\label{eq:main-family} \varphi_{r,e}(x)=x^q+qp^r x^{q+1}=x^{p^e}+p^{r+e}x^{p^e+1}, \qquad q=p^e,\tag{1}\] where \(r\ge 0\), together with its inverse Böttcher coordinate \[f_{r,e}(x)=x\sum_{k\ge 0}\frac{a_k(r,e)}{k!}x^k, \qquad a_0(r,e)=1,\] normalized by \[\varphi_{r,e}\bigl(f_{r,e}(x)\bigr)=f_{r,e}(x^q).\] When \(e\) is fixed we suppress it from the notation and write \(a_k(r)\) and \(f_r\). Equivalently, if \(\Phi_{r,e}=f_{r,e}^{-1}\), then \[\Phi_{r,e}\bigl(\varphi_{r,e}(x)\bigr)=\Phi_{r,e}(x)^q.\]
Our aim is to describe the coefficients \(a_k(r,e)\) directly in degree \(p^e\). In the special fiber \(r=0\) we determine all coefficients modulo \(p\) by a closed digit-sum law. In the higher fibers \(r\ge1\) we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a pure-power recursion with lag \(e\), and an exact radius formula. We formulate the degree-\(p^e\) problem first and view the degree-\(p^2\) case only as the specialization \(e=2\).
The case \(e=2\) is the family isolated by Salerno and Silverman in their wild conjectures [17]. Fu and Nie proved the radius in a substantially broader wild superattracting setting [18]. In their notation the Böttcher coordinate \(\Phi\) is our \(f_{r,2}^{-1}\), so the radius of \(\Phi^{-1}\) in [18] is exactly the radius of our \(f_{r,2}\). The formulas proved below for \(e>2\) belong to the present extension.
Why the degree-\(p^e\) theorem is not formal. The higher-fiber theorem is driven by the pure-power recursion. For general \(e\) this recursion has lag \(e\), and its branch word is \[(B^{e-1}A)^sB^\infty, \qquad s=\left\lceil\frac{r}{e}\right\rceil.\] This branch word is what produces both the stable pure-power slope and the exponent in the radius formula. Only after the general pattern has been established does one recover the degree-\(p^2\) case by setting \(e=2\).
Method. Our higher-fiber argument is built around a filtered cumulant principle. We write the \(B\)-coefficient valuation as a sum of two carry defects. In the divisible non-pure sector we then show that every unit scalar term is already carry-free, and from this point the degree-\(\Lambda_{r,e}\) initial unit sector is read off from the carry-free cumulant coefficient. This is the key step behind the leading monomial theorem on divisible classes. At stable layers one may still use the conceptual quotient with relations \(Y_j^p=Y_{j+1}\), but the scalar calculation is finished before passing to that quotient.
The clean-family results are the main theorems of the paper. After stating them, we turn to a perturbative extension showing that the same coefficient-level structure remains stable under sufficiently small higher-order tails. In the higher-fiber statement, the condition \[\mathop{\mathrm{ord}}_p(\vartheta_h)\ge \Lambda_{r,e}(h+1)+1\] is chosen so that every tail contribution lies strictly above the initial \(\Lambda_{r,e}\)-graded term.
We first state the special-fiber congruence law. In this statement \(a_k=a_k(0,e)\).
Theorem 1 (Digit-sum formula in the special fiber). Let \(k\ge 1\) and write \[k=a+d_1p+d_2p^2+\cdots+d_Np^N, \qquad 0\le a,d_i\le p-1,\] with \[s:=d_1+\cdots+d_N.\] Then \[\label{eq:digit-formula-aa} a_k\equiv (-1)^s(a+1)^s a_a \pmod p.\qquad{(1)}\] In fact, \[\label{eq:closed-digit-formula} a_k\equiv (-1)^{a+s}(a+1)^{a+s-1}\pmod p.\qquad{(2)}\]
As a first corollary, one obtains explicit congruences in the three residue classes \(0,-1,-2\pmod p\).
Theorem 2 (Special-fiber residue classes). For every integer \(m\ge 1\) one has \[a_{pm}\equiv (-1)^m, \qquad a_{pm-1}\equiv 0, \qquad a_{pm-2}\equiv -1\pmod p.\]
We then state the recursive theorem for the higher fibers. Fix \(r\ge 1\). Put \[v_i(r,e):=\mathop{\mathrm{ord}}_p\bigl(a_{p^i}(r,e)\bigr)\qquad (i\ge 0),\] and, when \(e\) is fixed, write simply \(v_i(r)\). For \(k=\sum_{i\ge 0}k_ip^i\), define \[\Lambda_{r,e}(k):=\sum_{i\ge 0}k_iv_i(r,e), \qquad m_{r,e}(k):=\prod_{i\ge 0}a_{p^i}(r,e)^{k_i}.\] Set \[\mu_e:=\frac{p^e-1}{p-1}=1+p+\cdots+p^{e-1}.\] For \(n\ge e\) define \[\begin{align} \Delta_{n,e}&:=\mu_e p^{n-e}-e,\\ A_{n,e}(r)&:=\Delta_{n,e}+v_{n-e}(r,e),\\ B_{n,e}(r)&:=p\,v_{n-1}(r,e), \end{align}\] and for \(n<e\) regard \(A_{n,e}\) as absent. Finally set \[\begin{align} \alpha_{n,e}&:=\frac{(p^n)!}{p^e(p^{n-e})!} && (n\ge e),\\ \gamma_{n,e}&:=\frac{(p^n)!}{p\bigl((p^{n-1})!\bigr)^p}\binom{p^e-1}{p-1} && (n\ge 1). \end{align}\]
Theorem 3 (Recursive structure in degree \(p^e\)). For every fixed \(r\ge 1\) and \(e\ge 2\), the following assertions hold.
**Global digit-weight lower bound.* For every \(k\ge 1\), \[\mathop{\mathrm{ord}}_p\bigl(a_k(r,e)\bigr)\ge \Lambda_{r,e}(k).\]*
**Leading monomial on divisible non-pure classes.* If \(p\mid k\) and \(k\) is not a power of \(p\), then \[a_k(r,e)\equiv m_{r,e}(k)\pmod {p^{\Lambda_{r,e}(k)+1}}.\]*
**Pure-power recursion.* One has \(v_0(r,e)=r\) and, for \(1\le n<e\), \[v_n(r,e)=p\,v_{n-1}(r,e)=p^nr.\] For every \(n\ge e\), \[v_n(r,e)=\min\{A_{n,e}(r),B_{n,e}(r)\}, \qquad A_{n,e}(r)\ne B_{n,e}(r).\] Moreover the corresponding leading branch is \[a_{p^n}(r,e)= \begin{cases} -\gamma_{n,e}a_{p^{n-1}}(r,e)^p+O\bigl(p^{v_n(r,e)+1}\bigr),&1\le n<e,\\[1mm] \alpha_{n,e}a_{p^{n-e}}(r,e)+O\bigl(p^{v_n(r,e)+1}\bigr),& n\ge e,\;A_{n,e}(r)<B_{n,e}(r),\\[1mm] -\gamma_{n,e}a_{p^{n-1}}(r,e)^p+O\bigl(p^{v_n(r,e)+1}\bigr),& n\ge e,\;B_{n,e}(r)<A_{n,e}(r). \end{cases}\] In particular \(v_n(r,e)\le p\,v_{n-1}(r,e)\) for all \(n\ge 1\).*
**Subadditivity of the digit weight.* For every finite sum \(N=n_1+\cdots+n_t\) of nonnegative integers, \[\Lambda_{r,e}(N)\le \Lambda_{r,e}(n_1)+\cdots+\Lambda_{r,e}(n_t).\]*
The following theorem gives the explicit branch pattern and the radius. Put \[s_{r,e}:=\left\lceil\frac{r}{e}\right\rceil, \qquad \lambda_{r,e}:=p^{-e s_{r,e}} \left(r+\frac{p^{e s_{r,e}}-1}{p-1}-e\,s_{r,e}\right),\] and \[\theta_{r,e}:=\frac{1}{p-1}-\lambda_{r,e} =p^{-e s_{r,e}}\left(\frac{1}{p-1}+e\,s_{r,e}-r\right).\] In particular, \(\theta_{r,e}>0\), since \(e\,s_{r,e}-r\ge0\). For \(n\ge 0\) define \[N_{r,e}(n):=\min\left(\left\lfloor\frac{n}{e}\right\rfloor,s_{r,e}\right), \qquad \varepsilon_{r,e}(n):=(-1)^{1+eN_{r,e}(n)}\in\{\pm1\}.\] Here \(N_{r,e}(n)\) is the number of \(A\)-branches encountered up to level \(n\).
Theorem 4 (Valuations and radius in degree \(p^e\)). Fix \(r\ge 1\) and \(e\ge 2\).
The pure-power branch word is \[(B^{e-1}A)^{s_{r,e}}B^\infty.\] Equivalently, for \(0\le j\le s_{r,e}\) and \(0\le t\le e-1\), \[v_{je+t}(r,e)=p^t\left(r+\frac{p^{j e}-1}{p-1}-e\,j\right),\] and for all \(n\ge es_{r,e}\), \[v_n(r,e)=\lambda_{r,e}p^n.\]
For every \(n\ge 0\), \[p^{-v_n(r,e)}a_{p^n}(r,e)\equiv \varepsilon_{r,e}(n)\pmod p.\] Consequently, if \(m=\sum_{i\ge 0}m_ip^i\), then \[a_{pm}(r,e)= \left(\prod_{i\ge 0}\varepsilon_{r,e}(i+1)^{m_i}\right) p^{\sum_{i\ge 0}m_i v_{i+1}(r,e)} +O\!\left(p^{\sum_{i\ge 0}m_i v_{i+1}(r,e)+1}\right).\]
If \(p\mid k\), then \[\mathop{\mathrm{ord}}_p\bigl(a_k(r,e)\bigr)=\Lambda_{r,e}(k)=\lambda_{r,e}k+O(1).\]
The \(p\)-adic radius of convergence of \(f_{r,e}\) is \[\rho(f_{r,e})=p^{-\theta_{r,e}} =p^{-\left(p^{-e s_{r,e}}\left(\frac{1}{p-1}+e\,s_{r,e}-r\right)\right)}.\] For \(e=2\), this is \(p^{-\frac{p^{-r}}{p-1}}\).
Theorem 5 (Tail-stable extension). Let \(\vartheta_h\in\mathbb{Q}_p\) for all \(h\ge1\), and consider the formal germ \[\widetilde{\varphi}_{r,e}(x)=x^q+qp^r x^{q+1}+q\sum_{h\ge1}\vartheta_hx^{q+1+h}, \qquad q=p^e,\] with inverse Böttcher coordinate \[\widetilde{f}_{r,e}(x)=x\sum_{k\ge0}\frac{\widetilde{a}_k(r,e)}{k!}x^k, \qquad \widetilde{\varphi}_{r,e}\bigl(\widetilde{f}_{r,e}(x)\bigr)=\widetilde{f}_{r,e}(x^q).\] Since the coefficient recursion at a fixed degree involves only finitely many \(h\), all statements below are understood coefficient-wise.
If \(r=0\) and \(\vartheta_h\in p\mathbb{Z}_p\) for all \(h\ge1\), then \[\widetilde{a}_k(0,e)\equiv a_k(0,e)\pmod p \qquad (k\ge0).\] In particular, the conclusions of Theorems 1 and 2 remain valid for \(\widetilde{a}_k(0,e)\).
Assume \(r\ge1\) and \[\mathop{\mathrm{ord}}_p(\vartheta_h)\ge \Lambda_{r,e}(h+1)+1 \qquad (h\ge1).\] For \(k=\sum_{i\ge0}k_ip^i\), set \[\widetilde{m}_{r,e}(k):=\prod_{i\ge0}\widetilde{a}_{p^i}(r,e)^{k_i}.\] Here \(\Lambda_{r,e}\) is the clean-family digit weight from Theorem 3. Then the analogues of Theorems 3 and 4 hold for the perturbed coefficients \(\widetilde{a}_k(r,e)\), with the same digit weight \(\Lambda_{r,e}\) and with \(\widetilde{m}_{r,e}(k)\) in place of \(m_{r,e}(k)\). In particular, the pure-power branch word, the normalized pure-power units, the valuation asymptotic, and the radius \(p^{-\theta_{r,e}}\) are unchanged.
Remark 1 (The degree-\(p^2\) specialization). When \(e=2\), the family 1 becomes \[\varphi_{r,2}(x)=x^{p^2}+p^{r+2}x^{p^2+1},\] which is exactly the wild degree-\(p^2\) family studied by Salerno–Silverman. In this specialization, Theorem 2 proves the special-fiber residue-class prediction of Salerno–Silverman, while Theorem 4 proves the higher-fiber valuation and radius prediction in the same family. The stronger results Theorems 1 and 3 may be viewed as coefficient-level refinements of those two predictions.
Although Theorem 5 is far from the full generality of [18], it shows that the coefficient-level structure developed here is not confined to the exact one-parameter model 1 : the special-fiber digit-sum law and the higher-fiber leading-term calculus persist under controlled higher-order perturbations.
Throughout the paper, if \(X,Y\in\mathbb{Q}_p\) and \(N\in\mathbb{Z}\), we write \[X=Y+O(p^N)\] to mean \(\mathop{\mathrm{ord}}_p(X-Y)\ge N\). All congruences modulo powers of \(p\) are applied only after the relevant quantities have been shown to be \(p\)-integral.
The paper is arranged as follows. Sections 2–6 deal with the special fiber. We prove the \(A\)–\(B\)–\(C\) recursion, the carry-defect integrality of the \(B\)-term, the residue class \(a=0\), and the vector-partition cumulant collapse for \(a\ne0\). Sections 7–8 deal with the higher fibers. We first set up the generalized recursion and the filtered cumulant argument, and then we prove Theorem 3, the branch word, and Theorem 4. Section 9 proves the tail-stable extension stated in Theorem 5.
We restrict throughout to odd primes. The case \(p=2\) brings in additional low-characteristic coincidences, and for this reason we do not discuss it here.
In this section we work in the special fiber \(r=0\). Thus \[\varphi(x)=x^q+qx^{q+1},\qquad q=p^e,\] and we write \(a_k=a_k(0,e)\). Since \(\varphi(x)\) is of the form \(x^m+mx^{m+1}R[\![x]\!]\) with \(m=q\) and \(R=\mathbb{Z}_p\), the coefficient-integrality theorem of Salerno–Silverman, namely [17], gives \(a_k\in\mathbb{Z}_p\) for all \(k\). Therefore all reductions modulo \(p\) in Sections 2–6 are legitimate.
The coefficient comparison gives \[\label{eq:ABC} a_k=A_k[x^k]-B_k[x^k]-C_k[x^k],\tag{2}\] where \[A_k:=\frac{k!}{q}\sum_{\ell=0}^{k-1}\frac{a_\ell}{\ell!}x^{q\ell},\qquad B_k:=\frac{k!}{q}\left(\sum_{\ell=0}^{k-1}\frac{a_\ell}{\ell!}x^\ell\right)^q,\] \[C_k:=k!x\left(\sum_{\ell=0}^{k-1}\frac{a_\ell}{\ell!}x^\ell\right)^{q+1}.\] Here \(P[x^n]\) denotes the coefficient of \(x^n\) in \(P(x)\).
For \(N=\sum_iN_ip^i\), \(0\le N_i\le p-1\), put \[S_p(N):=\sum_iN_i, \qquad T_p(N):=\frac{N!}{p^{\mathop{\mathrm{ord}}_p(N!)}},\] so that \(T_p(N)\) is the prime-to-\(p\) part of \(N!\). We repeatedly use Legendre’s formula \[\label{eq:legendre} \mathop{\mathrm{ord}}_p(N!)=\frac{N-S_p(N)}{p-1}\tag{3}\] and digit-sum subadditivity \[\label{eq:digit-subadditivity} S_p(u+v)\le S_p(u)+S_p(v).\tag{4}\]
Proposition 1 (The \(A\)-term vanishes modulo \(p\)). For every \(k\ge 1\), \[A_k[x^k]\equiv 0\pmod p.\]
Proof. If \(q\nmid k\), then \(A_k[x^k]=0\). If \(k=qn\), then \[A_k[x^k]=\frac{(qn)!}{q\,n!}a_n.\] Since \(S_p(qn)=S_p(n)\), Legendre’s formula gives \[\begin{align} \mathop{\mathrm{ord}}_p\left(\frac{(qn)!}{q\,n!}\right) &=\frac{qn-S_p(qn)}{p-1}-e-\frac{n-S_p(n)}{p-1} \\ &=\frac{(q-1)n}{p-1}-e =\mu_en-e. \end{align}\] Here \(\mu_e=(q-1)/(p-1)\ge e+1\) for \(p\ge3\) and \(e\ge2\), so the last number is at least \(1\). Hence the \(A\)-term is divisible by \(p\). ◻
Let \(a^\eta=\prod_{n\ge0}a_n^{\eta_n}\) be a monomial occurring in \(B_k[x^k]\). Then \[\sum_n\eta_n=q, \qquad \sum_n n\eta_n=k,\] and its scalar coefficient is \[\gamma_B(\eta)=\frac{k!}{q}\binom{q}{\eta_0,\eta_1,\ldots}\prod_{n\ge0}\frac{1}{(n!)^{\eta_n}}.\] Legendre’s formula gives \[\label{eq:B-valuation-general} (p-1)\mathop{\mathrm{ord}}_p(\gamma_B(\eta)) =-S_p(k)-1-e(p-1)+\sum_nS_p(\eta_n)+\sum_n\eta_nS_p(n).\tag{5}\] Equivalently, with \[c(\eta):=\frac{\sum_n\eta_nS_p(n)-S_p(k)}{p-1}, \qquad d(\eta):=\frac{\sum_nS_p(\eta_n)-1}{p-1},\] one has \[\label{eq:B-cd} \mathop{\mathrm{ord}}_p(\gamma_B(\eta))=c(\eta)+d(\eta)-e.\tag{6}\] The integers \(c(\eta)\) and \(d(\eta)\) are the carry defects in the weighted addition \(\sum n\eta_n=k\) and in the multiplicity addition \(\sum\eta_n=q\).
Lemma 1 (Carry depth for multiplicities). Let \(\eta_n\ge0\) and \(\sum_n\eta_n=p^e\). If \(d(\eta)=t<e\), then every \(\eta_n\) is divisible by \(p^{e-t}\). If \(t\ge e\), the assertion is vacuous.
Proof. If \(t\ge e\), there is nothing to prove. Thus assume \(t<e\). Let \[s_0:=\min_{\eta_n>0}\mathop{\mathrm{ord}}_p(\eta_n),\] and write \(\eta_n=p^{s_0}u_n\) with \(u_n\in\mathbb{Z}_{\ge0}\). Then at least one \(u_n\) is not divisible by \(p\), and \(\sum_nu_n=p^{e-s_0}\). The addition of the \(u_n\) to obtain \(p^{e-s_0}\) has at least one nonzero units digit among the summands; hence it must carry from the units place. Since the final number has all lower \(e-s_0\) digits equal to zero, carries must propagate through levels \(0,1,\ldots,e-s_0-1\). Thus the number of carries is at least \(e-s_0\). But this number of carries is \[\frac{\sum_nS_p(u_n)-1}{p-1} =\frac{\sum_nS_p(\eta_n)-1}{p-1}=d(\eta)=t.\] Therefore \(s_0\ge e-t\). ◻
Lemma 2 (Global \(B\)-coefficient integrality). For every \(k\ge1\) and every monomial coefficient \(\gamma_B(\eta)\) occurring in \(B_k[x^k]\), \[c(\eta)+d(\eta)\ge e.\] Equivalently, \(\gamma_B(\eta)\in\mathbb{Z}_p\).
Proof. Put \(t=d(\eta)\) and \(s=e-t\). If \(s\le0\), there is nothing to prove. By Lemma 1, \(\eta_n=p^s u_n\) for all \(n\). Let \[M:=\sum_nnu_n, \qquad A:=\sum_nu_nS_p(n).\] Then \(k=p^sM\) and \(S_p(k)=S_p(M)\). Digit-sum subadditivity gives \(S_p(M)\le A\), and since \(k\ge1\) we have \(A\ge1\). Hence \[\begin{align} c(\eta) &=\frac{p^sA-S_p(M)}{p-1} \\ &\ge \frac{(p^s-1)A}{p-1} \ge \frac{p^s-1}{p-1} \ge s=e-t. \end{align}\] Thus \(c(\eta)+d(\eta)\ge e\). ◻
Proposition 2 (Unit \(B\)-terms on divisible classes). Assume \(p\mid k\) and let a monomial in \(B_k[x^k]\) have \(p\)-adic unit scalar coefficient. If some occurring positive index is not divisible by \(p\), then necessarily \(k=p\) and the monomial is \[a_0^{q-p}a_1^p.\] Consequently, if \(p\mid k\) and \(k>p\), every unit-coefficient monomial in \(B_k[x^k]\) uses only positive indices divisible by \(p\).
Proof. The unit condition is \(c(\eta)+d(\eta)=e\). Suppose that some occurring positive index is not divisible by \(p\). If \(d(\eta)=e\), then \(c(\eta)=0\), so the addition \(\sum n\eta_n=k\) is carry-free. Since \(p\mid k\), its units digit is zero, and carry-freeness forces every occurring positive index to have units digit zero, a contradiction. Hence \(d(\eta)<e\).
Put \(s=e-d(\eta)\ge1\). By Lemma 1, \(\eta_n=p^s u_n\). Let \(M=\sum_nnu_n\) and \(A=\sum_nu_nS_p(n)\). Since \(c(\eta)=s\), we have \[p^sA-S_p(M)=s(p-1).\] As before, \(S_p(M)\le A\), and hence \[(p^s-1)A\le s(p-1).\] For \(s\ge2\) this is impossible, because \(p^s-1>s(p-1)\) for odd \(p\). Therefore \(s=1\), and then the inequality forces \(A=1\) and \(S_p(M)=1\). Thus exactly one positive index occurs after division by \(p\), with multiplicity one and digit sum one. Since we assumed an occurring positive index not divisible by \(p\), this index must be \(1\). Hence \(k=p\) and the corresponding multiplicities are \(\eta_1=p\), \(\eta_0=q-p\), as claimed. ◻
Proposition 3 (A global \(C\)-term lemma). For every \(k\ge1\), every monomial coefficient of \(C_k[x^k]\) is \(p\)-integral. Moreover:
if \(p\mid k\), then every monomial coefficient is divisible by \(p\), so \(C_k[x^k]\equiv0\pmod p\);
if \(k\equiv a\pmod p\) with \(1\le a\le p-1\), then the unique surviving term modulo \(p\) is \(a\,a_{k-1}\), i.e. \[C_k[x^k]\equiv a\,a_{k-1}\pmod p.\]
Proof. Let \(a^\eta=\prod_{n=0}^{k-1}a_n^{\eta_n}\) occur in \(C_k[x^k]\). Then \[\sum_{n=0}^{k-1}\eta_n=q+1, \qquad \sum_{n=0}^{k-1}n\eta_n=k-1,\] and \[\gamma_C(\eta)=k!\binom{q+1}{\eta_0,\ldots,\eta_{k-1}}\prod_{n=0}^{k-1}\frac{1}{(n!)^{\eta_n}}.\] A Legendre calculation gives \[\label{eq:C-valuation} (p-1)\mathop{\mathrm{ord}}_p(\gamma_C(\eta))=-S_p(k)-1+\Sigma(\eta)+T(\eta),\tag{7}\] where \[\Sigma(\eta):=\sum_{n=0}^{k-1}S_p(\eta_n), \qquad T(\eta):=\sum_{n=0}^{k-1}\eta_nS_p(n).\] Since \(S_p(q+1)=2\), digit-sum subadditivity gives \(\Sigma(\eta)\ge2\) and \(T(\eta)\ge S_p(k-1)\).
If \(p\mid k\), write \(k=p^\nu u\) with \(\nu\ge1\) and \(p\nmid u\). Then \(S_p(k-1)=S_p(k)-1+\nu(p-1)\), so 7 gives \[(p-1)\mathop{\mathrm{ord}}_p(\gamma_C(\eta))\ge\nu(p-1),\] and every \(C\)-coefficient is divisible by \(p\).
Assume now that \(k\equiv a\not\equiv0\pmod p\). Then \(S_p(k-1)=S_p(k)-1\), so all coefficients are \(p\)-integral, and a coefficient survives modulo \(p\) only if \[\Sigma(\eta)=2, \qquad T(\eta)=S_p(k-1).\] The condition \(\Sigma(\eta)=2\) leaves only the patterns \(\eta_t=q+1\) or \(\eta_r=q\), \(\eta_s=1\). In the first case, if \(t>0\), then \[S_p(k-1)=S_p((q+1)t)\le S_p(qt)+S_p(t)=2S_p(t)<(q+1)S_p(t)=T(\eta),\] a contradiction. Hence \(t=0\), which only gives the immediate case \(k=1\). In the second case, if \(r>0\), then \[S_p(k-1)=S_p(qr+s)\le S_p(r)+S_p(s)<qS_p(r)+S_p(s)=T(\eta),\] again a contradiction. Thus \(r=0\), \(s=k-1\), and the unique surviving monomial is \(\eta_0=q\), \(\eta_{k-1}=1\). Its coefficient is \[k!\binom{q+1}{q,1}\frac{1}{(k-1)!}=k(q+1)\equiv k\equiv a\pmod p.\] ◻
Theorem 6 (The case \(a=0\)). For every integer \(M\ge1\), \[a_{pM}\equiv (-1)^M\pmod p.\]
We give the proof in the next section.
We prove Theorem 6. The same truncated exponential identity works for every \(e\ge2\).
Lemma 3 (A truncated-exponential coefficient identity). For \(M\ge1\), set \[S_{M-1}(y):=\sum_{i=0}^{M-1}\frac{(-1)^i}{p^ii!}y^i.\] Then \[[y^M]S_{M-1}(y)^q =\frac{(-q/p)^M}{M!}-q\frac{(-1)^M}{p^M M!}.\] Equivalently, \[[y^M]\left(\sum_{i=0}^{M-1}\frac{y^i}{p^ii!}\right)^q =\frac{p^{(e-1)M}}{M!}-\frac{q}{p^M M!}.\]
Proof. Write \(E(y)=e^{-y/p}\) and \[R_M(y):=\sum_{i\ge M}\frac{(-1)^i}{p^ii!}y^i.\] Then \(S_{M-1}=E-R_M\) and \[-R_M(y)=-\frac{(-1)^M}{p^MM!}y^M+O(y^{M+1}).\] In the expansion \((E-R_M)^q\), all terms with at least two copies of \(R_M\) have \(y\)-adic order \(>M\). Hence \[\begin{align} [y^M]S_{M-1}(y)^q &=[y^M]E(y)^q-q\frac{(-1)^M}{p^MM!} \\ &=\frac{(-q/p)^M}{M!}-q\frac{(-1)^M}{p^MM!}. \end{align}\] Replacing \(y\) by \(-y\) gives the second formula. ◻
Lemma 4. One has \[a_1=-1, \qquad a_p\equiv -1\pmod p.\]
Proof. Write \(f(x)=x+a_1x^2+O(x^3)\). Comparing the coefficient of \(x^{q+1}\) in \[f(x)^q+qf(x)^{q+1}=f(x^q)\] gives \(qa_1+q=0\), hence \(a_1=-1\).
For \(a_p\), Propositions 1 and 3 give \(a_p\equiv -B_p[x^p]\pmod p\). By Proposition 2, the only unit-coefficient monomial is \(a_0^{q-p}a_1^p\). Its scalar is \[\gamma_p=\frac{p!}{q}\binom{q}{q-p,p}=\frac{(q-1)!}{(q-p)!}=\prod_{j=1}^{p-1}(q-j) \equiv (-1)^{p-1}(p-1)!\equiv -1\pmod p\] by Wilson’s theorem. Therefore \[B_p[x^p]\equiv \gamma_pa_1^p\equiv (-1)(-1)^p\equiv 1\pmod p,\] and \(a_p\equiv-1\pmod p\). ◻
Proof of Theorem 6. We argue by induction on \(M\). The case \(M=1\) is Lemma 4. Assume \(M\ge2\) and \[a_{pi}\equiv (-1)^i\pmod p\qquad(1\le i<M).\] By Propositions 1 and 3, \[a_{pM}\equiv -B_{pM}[x^{pM}]\pmod p.\] Since \(pM>p\), Proposition 2 shows that only indices divisible by \(p\) can survive modulo \(p\). With \(y=x^p\) we get \[\label{eq:B-div-special} B_{pM}[x^{pM}] \equiv \frac{(pM)!}{q}[y^M] \left(\sum_{i=0}^{M-1}\frac{a_{pi}}{(pi)!}y^i\right)^q \pmod p.\tag{8}\] Put \[\nu_i:=\frac{(pi)!}{p^ii!}.\] Then \(\nu_i\in\mathbb{Z}_p^\times\) and \[\label{eq:nu-i} \nu_i=\prod_{r=1}^i\prod_{t=1}^{p-1}(p(r-1)+t)\equiv ((p-1)!)^i\equiv (-1)^i\pmod p.\tag{9}\] Thus \(a_{pi}\nu_i^{-1}\equiv1\pmod p\) for \(0\le i<M\). Write \(a_{pi}\nu_i^{-1}=1+pz_i\) with \(z_i\in\mathbb{Z}_p\) and define \[H_M(y):=\sum_{i=0}^{M-1}\frac{y^i}{p^ii!}, \qquad L_M(y):=\sum_{i=0}^{M-1}z_i\frac{y^i}{p^ii!}.\] Then \[\sum_{i=0}^{M-1}\frac{a_{pi}}{(pi)!}y^i=H_M(y)+pL_M(y).\] We claim that \[\label{eq:HM-replacement} \frac{p^MM!}{q}[y^M](H_M+pL_M)^q \equiv \frac{p^MM!}{q}[y^M]H_M^q\pmod p.\tag{10}\] Indeed, \[(H_M+pL_M)^q-H_M^q =\sum_{j=1}^q\binom qj p^jH_M^{q-j}L_M^j.\] Write \[H_M(y)=\sum_{i=0}^{M-1}h_i y^i, \qquad L_M(y)=\sum_{i=0}^{M-1}\ell_i y^i,\] where \[h_i=\frac{1}{p^ii!}, \qquad \ell_i=z_i\frac{1}{p^ii!}.\] A typical term in \(M![y^M]H_M^{q-j}L_M^j\) is \[\binom{q-j}{a_0,\ldots,a_{M-1}} \binom{j}{b_0,\ldots,b_{M-1}} \frac{M!}{\prod_{i=0}^{M-1}(i!)^{a_i+b_i}} \frac{\prod_{i=0}^{M-1}z_i^{\,b_i}}{p^{\sum_i i(a_i+b_i)}},\] where \[\sum_i a_i=q-j, \qquad \sum_i b_i=j, \qquad \sum_i i(a_i+b_i)=M.\] The multinomial factor \(M!/\prod_i (i!)^{a_i+b_i}\) is an integer, and the last displayed condition gives the exact \(p\)-power denominator \(p^M\). Hence \[M![y^M]H_M^{q-j}L_M^j\in p^{-M}\mathbb{Z}_p.\] Therefore it is enough to show \[\mathop{\mathrm{ord}}_p\left(\frac{p^j}{q}\binom qj\right)\ge1\qquad(1\le j\le q).\] Using \(\binom qj=\frac{q}{j}\binom{q-1}{j-1}\), we get \[\frac{p^j}{q}\binom qj=\frac{p^j}{j}\binom{q-1}{j-1},\] so \[\mathop{\mathrm{ord}}_p\left(\frac{p^j}{q}\binom qj\right)\ge j-\mathop{\mathrm{ord}}_p(j)\ge1.\] This gives 10 .
Combining 8 and 10 , and using \((pM)!=p^MM!\nu_M\), gives \[a_{pM}\equiv -\nu_M\frac{p^MM!}{q}[y^M]H_M(y)^q\pmod p.\] By Lemma 3, \[[y^M]H_M(y)^q=\frac{p^{(e-1)M}}{M!}-\frac{q}{p^MM!}.\] Therefore \[a_{pM}\equiv -\nu_M\bigl(p^{e(M-1)}-1\bigr)\pmod p.\] Since \(M\ge2\) and \(e\ge2\), this is congruent to \(\nu_M\). Finally 9 gives \(\nu_M\equiv(-1)^M\pmod p\). ◻
The higher base-\(p\) digits of the index will be encoded by vectors.
Definition 1. Fix \(L\ge1\). A digit vector is an element \[d=(d_1,\ldots,d_L)\in\{0,1,\ldots,p-1\}^L.\] Its weight and numeric value are \[|d|:=d_1+\cdots+d_L, \qquad N(d):=\sum_{i=1}^Ld_ip^i.\] A vector partition of \(d\) is a formal product \[\Pi=\prod_{\beta\ne0}\beta^{r_\beta}, \qquad r_\beta\in\mathbb{Z}_{\ge0}, \qquad \sum_{\beta\ne0}r_\beta\beta=d.\] The total number of blocks is \(R(\Pi):=\sum_{\beta\ne0}r_\beta\).
Fix \(a\in\{0,1,\ldots,p-1\}\). For a digit vector \(\beta\) define \[H_\beta(y):=\sum_{j=0}^a\frac{a_{N(\beta)+j}}{j!}y^j, \quad F_a(y):=\sum_{j=0}^a\frac{a_j}{j!}y^j, \quad \widetilde{H}_d(y):=\sum_{j=0}^{a-1}\frac{a_{N(d)+j}}{j!}y^j.\] Write \[d!:=\prod_{i=1}^Ld_i!, \qquad \beta!:=\prod_{i=1}^L\beta_i!.\]
For an integer parameter \(t\) and \(N\ge1\), set \[\label{eq:Xi-def} \Xi_t(N):=\prod_{u=1}^{N-1}(t-u).\tag{11}\] In particular, when \(t=q=p^e\) one has \[\label{eq:Xi-q-modp} \Xi_q(N)=\Xi_{p^e}(N)\equiv (-1)^{N-1}(N-1)!\pmod p.\tag{12}\] If \(N\le q\), then \(\Xi_q(N)=\frac{(q-1)!}{(q-N)!}\). Hence if \(N>q\), then any formal vector-partition contribution with \(N\) positive blocks is \(0\pmod p\): no actual multinomial term occurs, because the zero-index multiplicity would be negative, and 12 shows that the formal continuation is divisible by \(p\). Moreover, in a vector partition of a fixed digit vector \(d\), every contributing block multiplicity \(r_\beta\) satisfies \(0\le r_\beta\le p-1\); since each digit of every block \(\beta\) also lies in \(\{0,\ldots,p-1\}\), the factorials \(r_\beta!\) and \((\beta!)^{r_\beta}\) are \(p\)-adic units. Later, when the zero-vector multiplicities \(n_j\) appear, one has \(\sum_{j=1}^a jn_j\le a\le p-1\), hence each \(n_j\le p-1\) and each \(n_j!\) is also a \(p\)-adic unit. Thus the \(p\)-divisibility of the formal factor \(\Xi_q(N)\) cannot be cancelled by any denominator.
Proposition 4 (Vector-partition expansion of the \(B\)-term). Let \(d\ne0\) be a digit vector and let \(k=N(d)+a\) with \(1\le a\le p-1\). Then \[\label{eq:vector-B} \begin{align} B_k[x^k]\equiv a![y^a]\Biggl( &\frac{\widetilde{H}_d(y)}{F_a(y)} +\sum_{\substack{\Pi\vdash d\\ \Pi\ne(d)}} (-1)^{R(\Pi)-1} \frac{(R(\Pi)-1)!\,d!}{\prod_{\beta\ne0}r_\beta!(\beta!)^{r_\beta}} \\ &\qquad\qquad\times \prod_{\beta\ne0}\left(\frac{H_\beta(y)}{F_a(y)}\right)^{r_\beta} \Biggr)\pmod p. \end{align}\qquad{(3)}\] The case \(d=0\) is handled separately in the proof of Theorem 1.
Proof. Let \(a^\eta=\prod a_n^{\eta_n}\) be a monomial in \(B_k[x^k]\). Since \(k\not\equiv0\pmod p\), a unit coefficient must satisfy \(c(\eta)+d(\eta)=e\). If \(d(\eta)<e\), then Lemma 1 forces every \(\eta_n\) to be divisible by \(p\), and then \(k=\sum n\eta_n\) is divisible by \(p\), impossible. Hence \[d(\eta)=e, \qquad c(\eta)=0.\] Thus every monomial surviving modulo \(p\) is weighted carry-free in the addition \(\sum n\eta_n=k\).
Write each positive index uniquely as \[n=N(\beta)+j, \qquad 0\le j\le a,\] where \(\beta\) records the digits in the \(p,p^2,\ldots,p^L\) positions. For \(\beta\ne0\) put \[m_{\beta,j}:=\eta_{N(\beta)+j}, \qquad r_\beta:=\sum_{j=0}^a m_{\beta,j},\] and for the zero vector put \(n_j:=\eta_j\) for \(1\le j\le a\). Carry-freeness is equivalent to \[\sum_{\beta\ne0}r_\beta\beta=d, \qquad \sum_{\beta\ne0}\sum_{j=0}^a jm_{\beta,j}+\sum_{j=1}^a jn_j=a.\] Thus the nonzero vectors form a vector partition \(\Pi=\prod\beta^{r_\beta}\) of \(d\). Since \[\sum_{j=1}^a jn_j\le a\le p-1,\] one has \(n_j\le p-1\) for every \(j\), so the zero-vector factorials \(n_j!\) are also \(p\)-adic units.
For \(\Pi=(d)\), only the top choice \(j=a\), namely the block \(N(d)+a=k\), is forbidden; the choices \(0\le j<a\) remain, which is exactly the replacement \(H_d\mapsto\widetilde{H}_d\). Fix \(\Pi\), and set \[R:=R(\Pi), \qquad t:=\sum_{j=1}^a n_j, \qquad M:=R+t.\] For an actual monomial one has \(M\le q\), so \(\eta_0=q-M\ge0\). For the formal extension to \(M>q\), we use 12 . Thus the same displayed scalar expression may be written uniformly as \[\begin{align} k!\frac{\Xi_q(M)}{\prod_{\beta,j}m_{\beta,j}!\prod_{j=1}^a n_j!} \prod_{\beta,j}\frac{1}{(N(\beta)+j)!^{m_{\beta,j}}} \prod_{j=1}^a\frac{1}{(j!)^{n_j}}. \end{align}\] Modulo \(p\), \[\Xi_q(M)=\prod_{u=1}^{M-1}(q-u)\equiv(-1)^{M-1}(M-1)!.\] By 12 , the same formal expression may be used uniformly even when \(M>q\). By Lucas’ theorem in multinomial form, applied to the carry-free addition of the positive indices, \[\frac{k!}{\prod_{\beta,j}(N(\beta)+j)!^{m_{\beta,j}}\prod_j(j!)^{n_j}} \equiv \frac{a!d!}{\prod_{\beta,j}j!^{m_{\beta,j}}(\beta!)^{m_{\beta,j}}\prod_j(j!)^{n_j}} \pmod p.\] Summing first over the zero-vector blocks gives the negative-binomial expansion of \(F_a(y)^{-R}\), and summing over \(m_{\beta,j}\) with fixed \(r_\beta\) gives \(H_\beta(y)^{r_\beta}/r_\beta!\). Collecting the factors gives exactly ?? . ◻
The first block of coefficients is independent of \(e\).
Proposition 5 (The first block modulo \(p\)). For \(0\le n\le p-1\), \[a_n\equiv (-1)^n(n+1)^{n-1}\pmod p,\] with the convention that the case \(n=0\) gives \(a_0=1\).
Proof. Let \(g(x)=\sum_{n\ge0}a_nx^n/n!\), so \(f(x)=xg(x)\) and \(g(0)=1\). The Böttcher equation is \[g(x^q)=g(x)^q(1+qxg(x)).\] Taking logarithms gives \[\log g(x^q)=q\log g(x)+\log(1+qxg(x)).\] For \(1\le n\le p-1\), the left side has zero \(x^n\)-coefficient. Divide the coefficient relation by \(q\). Since \[\frac{1}{q}\log(1+qxg(x))=xg(x)+\sum_{m\ge2}(-1)^{m-1}\frac{q^{m-1}}{m}x^mg(x)^m\] and \(m\le n<p\) in the relevant terms, all summands with \(m\ge2\) vanish modulo \(p\). Hence \[[x^n]\log g(x)\equiv-[x^{n-1}]g(x)\pmod p.\] Let \(F_{p-1}(x)=\sum_{n=0}^{p-1}a_nx^n/n!\). We get \[\label{eq:first-block-log} \log F_{p-1}(x)\equiv -xF_{p-1}(x)\pmod{(p,x^p)}.\tag{13}\] Set \(U=xF_{p-1}(x)\). Then \[U\equiv xe^{-U}\pmod{(p,x^{p+1})}.\] This congruence determines the coefficients of \(U\) recursively up to degree \(p\): the right-hand side has linear term \(x\), and the coefficient of \(x^m\) depends only on the lower coefficients of \(U\). Let \(\widetilde{U}\in x\mathbb{Q}_p[[x]]\) be the characteristic-zero solution of \[\widetilde{U}=xe^{-\widetilde{U}}.\] Then \(U\equiv \widetilde{U}\pmod{(p,x^{p+1})}\). By Lagrange inversion over \(\mathbb{Q}_p\) [19], [20], \[[x^m]\widetilde{U}=\frac{1}{m}[u^{m-1}]e^{-mu}=\frac{(-m)^{m-1}}{m!}\qquad(m\ge1).\] For \(1\le m\le p-1\) this is \(p\)-integral. For \(m=p\), the coefficient is \(p^{p-1}/p!=p^{p-2}/(p-1)!\in p\mathbb{Z}_p\), so it vanishes modulo \(p\). Dividing by \(x\) gives the desired formula for \(a_n/n!\) for \(0\le n\le p-1\). ◻
Corollary 1. For \(1\le a\le p-1\), \[\log F_a(y)\equiv -yF_a(y)\pmod{(p,y^{a+1})},\] and therefore \[a![y^a]\log F_a(y)\equiv -a\,a_{a-1}\pmod p.\]
Proof. Since \(F_a(y)-F_{p-1}(y)=O(y^{a+1})\), 13 gives the first congruence after truncation. Taking the \(y^a\) coefficient gives \[[y^a]\log F_a(y)\equiv-[y^{a-1}]F_a(y)=-\frac{a_{a-1}}{(a-1)!}\pmod p.\] ◻
Lemma 5 (Induction-to-block transfer). Assume that ?? is known for every index \(N(\beta)+j\) satisfying either \[0\le j<a\;\text{ and }\;|\beta|\le s, \qquad\text{or}\qquad j=a\;\text{ and }\;|\beta|<s.\] Then, for every nonzero digit vector \(\beta\) with \(|\beta|<s\), \[H_\beta(y)\equiv (-D)^{|\beta|}F_a(y)\pmod p, \qquad D:=1+y\frac{d}{dy}.\] Consequently, if \(d\) has weight \(s\), then \[\widetilde{H}_d(y)\equiv (-D)^sF_a(y)+(-1)^{s-1}(a+1)^s a_a\frac{y^a}{a!}\pmod p.\]
Proof. For \(|\beta|<s\), the displayed hypothesis gives \[a_{N(\beta)+j}\equiv (-1)^{|\beta|}(j+1)^{|\beta|}a_j\pmod p \qquad(0\le j\le a).\] Summing over \(j\) gives the first assertion because \(D^mF_a(y)=\sum_{j=0}^a(j+1)^ma_jy^j/j!\).
Now let \(d\) have weight \(s\). For \(0\le j<a\), the first part of the hypothesis applies with \(|d|=s\) and gives \[a_{N(d)+j}\equiv (-1)^s(j+1)^sa_j\pmod p.\] Summing only over \(0\le j<a\) therefore yields the displayed formula for \(\widetilde{H}_d(y)\); the missing top term \(j=a\) contributes exactly \[(-1)^{s-1}(a+1)^sa_a\frac{y^a}{a!}.\] ◻
For nonzero digit vectors define \[U_\beta(y):=(-1)^{|\beta|}\frac{D^{|\beta|}F_a(y)}{F_a(y)}.\] Introduce variables \(t=(t_1,\ldots,t_L)\) and write \(t^\beta=\prod_i t_i^{\beta_i}\). Define cumulants \(K_d(U)\) by \[\label{eq:cumulants} \log\left(1+\sum_{\beta\ne0}U_\beta(y)\frac{t^\beta}{\beta!}\right) =\sum_{d\ne0}K_d(U)\frac{t^d}{d!}.\tag{14}\]
Proposition 6 (Collapse to one variable). Let \(d\) be a nonzero digit vector and set \(s=|d|\). Then \[a![y^a]K_d(U)=(-1)^s a![y^a]N^s\log F_a(y), \qquad N:=y\frac{d}{dy}.\] Consequently, \[a![y^a]K_d(U)\equiv (-1)^{s+1}a^{s+1}a_{a-1}\pmod p.\]
Proof. To compute the coefficient of the fixed monomial \(t^d\), we may enlarge the block set from nonzero digit vectors \(\beta\in\{0,\ldots,p-1\}^L\) to all nonzero \(\beta\in\mathbb{Z}_{\ge0}^L\), because any term with some \(\beta_i>d_i\) cannot contribute to \(t^d\). Since \(0\le d_i\le p-1\), this enlargement does not change the coefficient under consideration.
After this harmless enlargement, \(U_\beta\) depends only on \(|\beta|\), so the series in 14 depends only on \(T=t_1+\cdots+t_L\): \[1+\sum_{\beta\ne0}U_\beta\frac{t^\beta}{\beta!} =\sum_{m\ge0}(-1)^m\frac{D^mF_a(y)}{F_a(y)}\frac{T^m}{m!} =\frac{e^{-TD}F_a(y)}{F_a(y)}.\] Since \(D=1+N\) and \(e^{-TN}F_a(y)=F_a(ye^{-T})\), the logarithm is \[-T+\log F_a(ye^{-T})-\log F_a(y).\] The coefficient of \(t^d/d!\) in a series depending only on \(T\) is the coefficient of \(T^{|d|}/|d|!\). Hence the first formula follows. The linear term \(-T\) has no \(y^a\) coefficient. Finally \([y^a]N^s\log F_a=a^s[y^a]\log F_a\), and Corollary 1 gives the congruence. ◻
Proof of ?? . The case \(a=0\) is Theorem 6. For \(1\le a\le p-1\) we argue by outer induction on \(a\) and inner induction on the higher digit weight \(s\).
Fix \(a\) and assume the theorem is known for smaller residue classes. If \(s=0\), then \(k=a\) and the claim is tautological. Let \(s\ge1\), let \(d\) be a digit vector of weight \(s\), and put \(k=N(d)+a\). By Proposition 3 and the already known formula in the residue class \(a-1\), \[\label{eq:C-in-main} C_k[x^k]\equiv a\,a_{k-1}\equiv (-1)^s a^{s+1}a_{a-1}\pmod p.\tag{15}\] By Proposition 1, \[\label{eq:A-in-main} A_k[x^k]\equiv0\pmod p.\tag{16}\]
For the \(B\)-term, put \(V_\beta(y)=H_\beta(y)/F_a(y)\), and let \(K_d(V)\) be defined by 14 with \(U_\beta\) replaced by \(V_\beta\). Since the one-block partition contributes \(V_d\), Proposition 4 gives \[\label{eq:B-cumulant-main} B_k[x^k]\equiv a![y^a]\left(\frac{\widetilde{H}_d(y)}{F_a(y)}+K_d(V)-V_d(y)\right)\pmod p.\tag{17}\] Every block in a proper partition of \(d\) has weight smaller than \(s\). Therefore Lemma 5, applied with the outer induction hypothesis on the residue class and the inner induction hypothesis on the higher digit weight, allows us to replace \(V_\beta\) by \(U_\beta\) in the proper-partition contribution: \[a-V_d)\equiv a-U_d)\pmod p.\] Moreover Lemma 5 gives \[a![y^a]\left(\frac{\widetilde{H}_d(y)}{F_a(y)}-U_d(y)\right) \equiv (-1)^{s-1}(a+1)^s a_a\pmod p.\] Together with Proposition 6, this yields \[\label{eq:B-main-special} B_k[x^k]\equiv (-1)^{s-1}(a+1)^sa_a+(-1)^{s+1}a^{s+1}a_{a-1}\pmod p.\tag{18}\] Substituting 15 , 16 , and 18 into 2 , we get \[\begin{align} a_k &\equiv-\bigl((-1)^{s-1}(a+1)^sa_a+(-1)^{s+1}a^{s+1}a_{a-1}\bigr) -(-1)^s a^{s+1}a_{a-1} \\ &\equiv (-1)^s(a+1)^sa_a\pmod p. \end{align}\] The two inductions are complete. ◻
Proof of ?? . If \(a=0\), then ?? gives \(a_k\equiv(-1)^s\pmod p\), which is ?? . If \(1\le a\le p-1\), then ?? and Proposition 5 give \[a_k\equiv (-1)^s(a+1)^sa_a \equiv (-1)^{s+a}(a+1)^{s+a-1}\pmod p.\] ◻
Proof of Theorem 2. The congruence \(a_{pm}\equiv(-1)^m\pmod p\) is Theorem 6. For the second congruence, write \[pm-1=(m-1)p+(p-1), \qquad s:=S_p(m-1).\] If \(m=1\), Proposition 5 gives \(a_{p-1}\equiv0\pmod p\). If \(m\ge2\), then \(s\ge1\), and ?? gives \[a_{pm-1}\equiv(-1)^sp^sa_{p-1}\equiv0\pmod p.\] For the third congruence, write \[pm-2=(m-1)p+(p-2), \qquad s:=S_p(m-1).\] Then ?? gives \[a_{pm-2}\equiv(-1)^s(p-1)^sa_{p-2}\equiv a_{p-2}\pmod p,\] and Proposition 5 gives \(a_{p-2}\equiv-1\pmod p\). ◻
Fix \(r\ge1\) and \(e\ge2\), and put \(q=p^e\). We now write \(a_k(r)=a_k(r,e)\). Since \[\varphi_{r,e}(x)=x^q+qp^rx^{q+1}\in x^q+qx^{q+1}\mathbb{Z}_p[\![x]\!],\] [17] gives \(a_k(r)\in\mathbb{Z}_p\) for every \(k\). Comparing the coefficient of \(x^{q+1}\) in the Böttcher equation \[\varphi_{r,e}\bigl(f_{r,e}(x)\bigr)=f_{r,e}(x^q)\] immediately gives \[\label{eq:a1-r} a_1(r)=-p^r,\tag{19}\] since the left-hand side contributes \(q a_1(r)+q p^r\) to \(x^{q+1}\), while the right-hand side has no \(x^{q+1}\) term. The recursion is \[\label{eq:ABC-r} a_k(r)=A_k[x^k]-B_k[x^k]-p^rC_k[x^k],\tag{20}\] where \(A_k,B_k,C_k\) are the same expressions as in Section 2, with \(a_\ell\) replaced by \(a_\ell(r)\). Thus all scalar coefficient formulas for \(A\), \(B\), and \(C\) remain valid. In particular, 6 becomes \[\mathop{\mathrm{ord}}_p(\gamma_B(\eta))=c(\eta)+d(\eta)-e,\] and all \(B\)-coefficients are \(p\)-integral by Lemma 2. If \(p\mid k\), every coefficient of \(C_k[x^k]\) is divisible by \(p\) by Proposition 3.
We begin with a low-index bound for the initial pure-power levels.
Lemma 6 (Low-index bound). For \(r\ge1\), \(e\ge2\), and \(0\le j\le p-1\), \[\mathop{\mathrm{ord}}_p\bigl(a_j(r)\bigr)\ge jr.\] Consequently, any product of coefficients whose lower indices have total sum \(N<p\) has valuation at least \(rN\).
Proof. Let \[g_r(x):=\sum_{j\ge0}\frac{a_j(r)}{j!}x^j,\] so that \(f_{r,e}(x)=xg_r(x)\) and \(g_r(0)=1\). The Böttcher equation becomes \[g_r(x^q)=g_r(x)^q\bigl(1+qp^r xg_r(x)\bigr),\qquad q=p^e.\] Taking logarithms gives \[\log g_r(x^q)=q\log g_r(x)+\log\bigl(1+qp^r xg_r(x)\bigr).\] Fix \(1\le n\le p-1\). The coefficient of \(x^n\) on the left side is \(0\). Dividing the coefficient relation by \(q\) therefore gives \[[x^n]\log g_r(x)=-p^r[x^{n-1}]g_r(x)+E_n,\] where \(E_n\) is a sum of terms coming from \[\sum_{m\ge2}(-1)^{m-1}\frac{q^{m-1}p^{mr}}{m}x^m g_r(x)^m.\] We prove the bound by induction on \(n\). The case \(n=1\) is 19 . Assume it for all smaller indices. A monomial contributing to the coefficient of \(x^{n-m}\) in \(g_r(x)^m\) is a product of lower coefficients whose total index is \(n-m\), so by the induction hypothesis it has valuation at least \(r(n-m)\). Since \(m\le n<p\), the denominator \(m\) is a \(p\)-adic unit. Therefore every summand of \(E_n\) has valuation at least \[(m-1)e+mr+r(n-m)=rn+(m-1)e\ge rn+2.\] On the other hand, the coefficient \([x^n]\log g_r(x)\) is \(a_n(r)/n!\) plus a polynomial in \(a_1(r),\ldots,a_{n-1}(r)\). Each monomial in that polynomial has total lower index \(n\), hence valuation at least \(rn\). Therefore the displayed relation gives \[\mathop{\mathrm{ord}}_p\!\left(\frac{a_n(r)}{n!}\right)\ge rn.\] Since \(n!\in\mathbb{Z}_p^\times\) for \(n<p\), the claim follows. ◻
Lemma 7 (The first two pure powers). For \(r\ge1\) and \(e\ge2\), \[v_0(r,e)=r, \qquad v_1(r,e)=pr,\] and \[p^{-v_i(r,e)}a_{p^i}(r,e)\equiv -1\pmod p \qquad (i=0,1).\]
Proof. The case \(i=0\) is 19 . For \(i=1\), take \(k=p\) in 20 . The \(A\)-term is \(0\) because \(q\nmid p\). By Proposition 3 and Lemma 6, the term \(p^rC_p[x^p]\) has valuation at least \(pr+1\): every monomial coefficient of \(C_p[x^p]\) is divisible by \(p\), and its monomials are products of lower coefficients with total lower index \(p-1\).
By Proposition 2, the unique unit-coefficient monomial in \(B_p[x^p]\) is \(a_0(r)^{q-p}a_1(r)^p\). Its scalar coefficient is \[\begin{align} \gamma_{1,e} &=\frac{p!}{q}\binom{q}{q-p,p} =\frac{(q-1)!}{(q-p)!} \\ &=\prod_{j=1}^{p-1}(q-j) \equiv (-1)^{p-1}(p-1)! \equiv -1\pmod p. \end{align}\] by Wilson’s theorem. All other \(B\)-monomials have scalar coefficient in \(p\mathbb{Z}_p\); by Lemma 6 their coefficient-products have valuation at least \(pr\), so they contribute only \(O(p^{pr+1})\). Therefore \[a_p(r)\equiv -\gamma_{1,e}a_1(r)^p\equiv -p^{pr}\pmod{p^{pr+1}}.\] This gives the case \(i=1\). ◻
Lemma 8. For every \(m\ge0\), \[T_p(p^m)\equiv (-1)^m\pmod p.\]
Proof. The assertion is clear for \(m=0\). For \(m\ge1\), group the integers from \(1\) to \(p^m\) according to their \(p\)-adic valuation. The prime-to-\(p\) parts of the numbers with valuation \(j\) contribute, modulo \(p\), the product of all nonzero residues modulo \(p\), repeated \(p^{m-j-1}\) times. Thus \[T_p(p^m)\equiv \prod_{j=0}^{m-1}((p-1)!)^{p^{m-j-1}} \equiv \prod_{j=0}^{m-1}(-1)^{p^{m-j-1}} \equiv (-1)^m\pmod p,\] using Wilson’s theorem and the fact that \(p\) is odd. ◻
We now isolate the pure-power part of the \(B\)-term.
Lemma 9 (Pure-power unit \(B\)-monomial). Let \(n\ge1\). In \(B_{p^n}[x^{p^n}]\), the unique monomial with \(p\)-adic unit scalar coefficient is \[a_0(r)^{q-p}a_{p^{n-1}}(r)^p.\] Its scalar coefficient is \(\gamma_{n,e}\) and satisfies \(\gamma_{n,e}\in\mathbb{Z}_p^\times\) and \(\gamma_{n,e}\equiv-1\pmod p\).
Proof. Let a unit-coefficient monomial have multiplicities \(\eta_i\). Then \(c(\eta)+d(\eta)=e\). If \(d(\eta)=e\), then \(c(\eta)=0\), so the weighted addition \(\sum_i i\eta_i=p^n\) is carry-free. Since the target is a pure power, the positive part would then consist of a single index \(p^n\) with multiplicity one, but every index occurring in \(B_{p^n}[x^{p^n}]\) is \(<p^n\). Hence \(d(\eta)<e\).
Put \(s=e-d(\eta)\ge1\). By Lemma 1, we may write \(\eta_i=p^s u_i\). Let \[M:=\sum_i i u_i, \qquad A:=\sum_i u_iS_p(i).\] Since \(c(\eta)=s\), equation 6 becomes \[p^sA-S_p(M)=s(p-1).\] Because \(S_p(M)\le A\), we obtain \[(p^s-1)A\le s(p-1).\] For \(s\ge2\) this is impossible, because \(p^s-1>s(p-1)\) for odd \(p\). Hence \(s=1\). The same inequality then forces \(A=1\) and \(S_p(M)=1\). Thus exactly one positive index occurs after division by \(p\), with multiplicity one and digit sum one. Since \[p^n=\sum_i i\eta_i=p\sum_i i u_i=pM,\] that index must be \(M=p^{n-1}\). Therefore \(\eta_{p^{n-1}}=p\) and \(\eta_0=q-p\).
The scalar coefficient is \[\gamma_{n,e} =\frac{(p^n)!}{p\bigl((p^{n-1})!\bigr)^p}\binom{p^e-1}{p-1}.\] The first factor is a \(p\)-adic unit, and by Lemma 8 \[\frac{(p^n)!}{p\bigl((p^{n-1})!\bigr)^p} \equiv \frac{T_p(p^n)}{T_p(p^{n-1})^p} \equiv \frac{(-1)^n}{((-1)^{n-1})^p} \equiv -1\pmod p.\] Lucas’ theorem gives \(\binom{p^e-1}{p-1}\equiv1\pmod p\), so \(\gamma_{n,e}\equiv-1\pmod p\). ◻
Proposition 7 (Exact decomposition on pure powers). For every \(n\ge1\) there exist polynomials \(R_n,S_n\in\mathbb{Z}_p[a_1(r),\ldots,a_{p^n-1}(r)]\) such that \[\label{eq:pure-decomp-small} a_{p^n}(r)=-\gamma_{n,e}a_{p^{n-1}}(r)^p+pR_n+p^{r+1}S_n \qquad(1\le n<e),\qquad{(4)}\] and \[\label{eq:pure-decomp-large} a_{p^n}(r)=\alpha_{n,e}a_{p^{n-e}}(r)-\gamma_{n,e}a_{p^{n-1}}(r)^p+pR_n+p^{r+1}S_n \qquad(n\ge e).\qquad{(5)}\] Moreover,
every monomial of \(R_n\) is a product of lower coefficients whose weighted total index is \(p^n\);
every monomial of \(S_n\) is a product of lower coefficients whose weighted total index is \(p^n-1\);
all scalar coefficients of \(R_n\) and \(S_n\) are \(p\)-integral;
\[\mathop{\mathrm{ord}}_p(\alpha_{n,e})=\Delta_{n,e}=\mu_ep^{n-e}-e, \qquad p^{-\Delta_{n,e}}\alpha_{n,e}\equiv(-1)^e\pmod p.\]
Proof. The \(A\)-term contributes only when \(n\ge e\), in which case \[A_{p^n}[x^{p^n}]=\frac{(p^n)!}{p^e(p^{n-e})!}a_{p^{n-e}}(r)=\alpha_{n,e}a_{p^{n-e}}(r).\] Legendre’s formula gives the stated valuation. The unit congruence follows from Lemma 8: \[p^{-\mathop{\mathrm{ord}}_p(\alpha_{n,e})}\alpha_{n,e} \equiv \frac{T_p(p^n)}{T_p(p^{n-e})} \equiv \frac{(-1)^n}{(-1)^{n-e}} \equiv (-1)^e\pmod p.\]
By Lemma 9, the unique \(B\)-monomial with unit scalar coefficient is \(a_0(r)^{q-p}a_{p^{n-1}}(r)^p\), and its scalar coefficient is \(\gamma_{n,e}\). Every other \(B\)-monomial has scalar coefficient in \(p\mathbb{Z}_p\) by Lemma 2. Hence the remaining \(B\)-contribution may be written as \(pR_n\), where every monomial of \(R_n\) has weighted total index \(p^n\).
Likewise, since \(p\mid p^n\), Proposition 3 gives \[C_{p^n}[x^{p^n}]\in p\mathbb{Z}_p[a_1(r),\ldots,a_{p^n-1}(r)].\] Every monomial occurring there has weighted total index \(p^n-1\), so after multiplying by the outer factor \(p^r\) in 20 we obtain the term \(p^{r+1}S_n\). Thus the stated support and integrality properties of \(R_n\) and \(S_n\) follow. ◻
We are ready to prove Theorem 3. Throughout this section \(r,e\) are fixed, and we write \(v_i=v_i(r,e)\), \(\Lambda=\Lambda_{r,e}\), and \(m=m_{r,e}\) when no confusion can arise. By the Section 7 formula 19 , one has \(v_0=r\).
Outline of the higher-fiber argument. We treat the divisible non-pure case in four steps. First, global \(B\)-coefficient integrality gives \(p\)-integral scalar coefficients. Second, the weight-excess identity controls the filtered degree under the pure-power inequalities already known. Third, every unit scalar term in the divisible non-pure sector is carry-free. Finally, the carry-free cumulant coefficient gives the degree-\(\Lambda\) initial unit sector. We use this scheme repeatedly below.
We refer to the four assertions of Theorem 3 as (L), (D), (P), and (S). For \(N\ge1\), let \(\mathcal{I}(N)\) be the package consisting of
(L) for every \(k\le N\);
(D) for every divisible non-pure \(k\le N\);
(P) for every pure power \(p^n\le N\) with \(n\ge1\);
(S) for every finite sum \(n_1+\cdots+n_t=M\) with \(M\le N\).
We prove \(\mathcal{I}(N)\) for all \(N\) by strong induction. In the step from \(\mathcal{I}(N-1)\) to \(\mathcal{I}(N)\), we first handle the pure-power case \(N=p^n\), then the divisible non-pure case, and finally the global lower bound at the general index \(N\). The subadditivity statement is recovered each time from the inequalities \(v_{j+1}\le p v_j\). In this way the dependence between (L), (D), (P), and (S) is explicit, and there is no circularity.
Two auxiliary lemmas will be used repeatedly.
Lemma 10 (Top-digit lower bound from lower levels). Assume that \[v_{i+1}\le p v_i\qquad (0\le i\le n-2).\] Let \(e_j\) be nonnegative integers with \(0\le j<p^n\) and \[\sum_{j=0}^{p^n-1}e_jj=p^n.\] Then \[\sum_{j=0}^{p^n-1}e_j\Lambda(j)\ge p\,v_{n-1}.\]
Proof. Write every \(j\) in base \(p\) and regard the left side as the total digit weight of the multiset consisting of \(e_j\) copies of \(j\). Passing from this multiset to the base-\(p\) representation of the total \(p^n\) amounts to repeatedly replacing \(p\) copies of \(p^i\) by one copy of \(p^{i+1}\). We perform all carries below level \(n-1\) but do not carry the resulting \(p\) copies of \(p^{n-1}\) to level \(n\). Each carry weakly decreases the total weight because \(v_{i+1}\le pv_i\). Therefore the initial weight is at least the final weight \(p v_{n-1}\). ◻
Lemma 11 (Weight excess from carries in degree \(p^e\)). Fix \(M\ge1\) and write \[M=M_0+M_1p+\cdots+M_sp^s,\qquad 0\le M_i\le p-1.\] Let \((e_i)_{0\le i<M}\) satisfy \[\sum_{i=0}^{M-1}e_i=p^e, \qquad \sum_{i=0}^{M-1}ie_i=M.\] Write \(i=\sum_{j\ge0}i_jp^j\) and set \[u_j:=\sum_{i=0}^{M-1}e_i i_j.\] Let \(c_j\ge0\) be the carry numbers determined by \[u_j=M_j+p c_j-c_{j-1}\qquad (j\ge0),\qquad c_{-1}:=0.\] Then \[\label{eq:weight-excess-pe} \sum_{j\ge0}u_jv_{j+1}-\Lambda(pM) =\sum_{j\ge0}c_j\bigl(pv_{j+1}-v_{j+2}\bigr).\qquad{(6)}\] If, in addition, \[v_{j+2}\le pv_{j+1} \qquad\text{for every }j\text{ with }c_j>0,\] then \[\sum_{j\ge0}u_jv_{j+1}\ge \Lambda(pM),\] and equality holds if and only if every nonzero carry \(c_j\) occurs at a level with \(v_{j+2}=pv_{j+1}\).
Proof. Since the base-\(p\) digits of \(pM\) are \(0,M_0,M_1,\ldots,M_s\), one has \[\Lambda(pM)=\sum_{j\ge0}M_jv_{j+1}.\] Subtracting this from \(\sum_j u_jv_{j+1}\) and using the defining relation for the carries gives \[\begin{align} \sum_{j\ge0}u_jv_{j+1}-\Lambda(pM) &=\sum_{j\ge0}(pc_j-c_{j-1})v_{j+1}\\ &=\sum_{j\ge0}c_j\bigl(pv_{j+1}-v_{j+2}\bigr), \end{align}\] which is ?? . Under the displayed hypothesis every summand on the right is nonnegative, and the final assertion is immediate. ◻
Lemma 12 (Carry-free coefficient in degree \(p^e\)). Let \(M\ge1\) be not a power of \(p\), and write \(M=M_0+M_1p+\cdots+M_sp^s\). Introduce variables \(Y_0,\ldots,Y_s\) and put \[Y^{[i]}:=\prod_jY_j^{i_j}\quad\text{if }i=\sum_ji_jp^j, \qquad Y^{[0]}:=1.\] Then the coefficient of \(Y^{[M]}\) in \[\frac{(pM)!}{p^e}[y^M] \left(\sum_{i=0}^{M-1}\frac{Y^{[i]}}{(pi)!}y^i\right)^{p^e}\] is congruent to \(-1\) modulo \(p\).
Proof. A contributing multi-index has multiplicities \(\eta_i\) satisfying \(\sum_i\eta_i=p^e\), \(\sum_i i\eta_i=M\), and the condition on the \(Y\)-exponent says that this addition is digitwise carry-free and has digit vector \((M_0,\ldots,M_s)\).
Set \(\nu_i=(pi)!/(p^ii!)\). As in 9 , \(\nu_i\equiv (-1)^i\pmod p\). Since \(\sum_i i\eta_i=M\), \[\frac{\nu_M}{\prod_i\nu_i^{\eta_i}} \equiv (-1)^{M-\sum_i i\eta_i} \equiv 1\pmod p,\] so the signs from \(\nu_M\) and the \(\nu_i\) cancel. For fixed nonzero block multiplicities, the remaining scalar is \[(-1)^{N-1}\frac{(N-1)!\,d!}{\prod_{\beta\ne0}r_\beta!(\beta!)^{r_\beta}},\] where \(d=(M_0,\ldots,M_s)\) and \(N=\sum_{\beta\ne0}r_\beta\). Indeed, the only place where \(e\) enters is the falling product \[\Xi_{p^e}(N)=\prod_{u=1}^{N-1}(p^e-u)\equiv(-1)^{N-1}(N-1)!\pmod p.\] By 12 , the carry-free vector-partition sum may be taken uniformly without imposing \(N\le q\). Thus the sum over all proper vector partitions of \(d\) is the cumulant coefficient in \[\log\left(1+\sum_{\beta\ne0}\frac{t^\beta}{\beta!}\right).\] To compute the coefficient of \(t^d\), we may temporarily enlarge the block set to all nonzero \(\beta\in\mathbb{Z}_{\ge0}^{s+1}\), because any coordinate \(\beta_j>d_j\) cannot contribute to \(t^d\). Then \[1+\sum_{\beta\ne0}\frac{t^\beta}{\beta!} =\prod_{j=0}^s\left(\sum_{n\ge0}\frac{t_j^n}{n!}\right) =e^{t_0+\cdots+t_s}.\] Hence the logarithm is \(t_0+\cdots+t_s\). Because \(M\) is not a power of \(p\), the digit vector \(d\) is not a standard basis vector, so the full cumulant coefficient is \(0\). The one-block partition of \(d\) is excluded by the range \(0\le i<M\). Therefore the proper-partition sum is \(-1\pmod p\). ◻
Lemma 13 (Unit terms are carry-free in the divisible non-pure sector). Let \(M\ge1\) be not a power of \(p\), and let \(\eta=(\eta_i)_{0\le i<M}\) satisfy \[\sum_{i=0}^{M-1}\eta_i=p^e, \qquad \sum_{i=0}^{M-1}i\eta_i=M.\] Consider the corresponding term in \[\frac{(pM)!}{p^e}[y^M] \left(\sum_{i=0}^{M-1}\frac{Y^{[i]}}{(pi)!}y^i\right)^{p^e}.\] If its scalar coefficient is a \(p\)-adic unit, then the addition \(\sum_i i\eta_i=M\) is carry-free. Equivalently, the associated monomial in the variables \(Y_j\) is already \(Y^{[M]}\).
Proof. By 6 , the unit condition is \(c(\eta)+d(\eta)=e\). Suppose first that \(c(\eta)>0\). Then \(d(\eta)<e\). Put \(s=e-d(\eta)\ge1\). By Lemma 1, every \(\eta_i\) is divisible by \(p^s\); write \(\eta_i=p^su_i\). Let \[M':=\sum_i iu_i, \qquad A:=\sum_i u_iS_p(i).\] Then \(M=p^sM'\) and the equality \(c(\eta)=s\) becomes \[p^sA-S_p(M')=s(p-1).\] Since \(S_p(M')\le A\), we get \[(p^s-1)A\le s(p-1).\] For \(s\ge2\) this is impossible. Therefore \(s=1\), and the same inequality forces \(A=1\) and \(S_p(M')=1\). Hence \(M'\) is a power of \(p\), and the unique positive index after division by \(p\) is a power of \(p\). Therefore \(M=pM'\) is itself a power of \(p\), contradicting the hypothesis. This contradiction shows that \(c(\eta)=0\).
Thus the addition \(\sum_i i\eta_i=M\) is carry-free. The exponent vector in the variables \(Y_j\) is therefore exactly the base-\(p\) digit vector of \(M\), so the monomial is \(Y^{[M]}\). ◻
In particular, every unit scalar term in the divisible non-pure sector is carry-free; consequently, the degree-\(\Lambda\) initial unit sector is computed by the carry-free cumulant coefficient.
For the associated-graded formalism, let \[\mathcal{R}_{r,e}:=\mathbb{Z}_p[Y_0,Y_1,\ldots], \qquad \operatorname{wt}(Y_j):=v_{j+1}(r,e).\] For an integer \(\lambda\in\mathbb{Z}_{\ge0}\), define \[F^\lambda\mathcal{R}_{r,e}:= \left\{\sum_{\nu} c_\nu Y^\nu: \mathop{\mathrm{ord}}_p(c_\nu)+\sum_{j\ge0}\nu_jv_{j+1}(r,e)\ge \lambda \text{ for every }\nu\right\}.\] Equivalently, \(F^\lambda\mathcal{R}_{r,e}\) is the \(\mathbb{Z}_p\)-span of the monomials \(p^aY^\nu\) with \[a+\sum_{j\ge0}\nu_jv_{j+1}(r,e)\ge \lambda.\] We write \[\operatorname{gr}_\Lambda\mathcal{R}_{r,e}:=\bigoplus_{\lambda\in\mathbb{Z}_{\ge0}}F^\lambda\mathcal{R}_{r,e}/F^{\lambda+1}\mathcal{R}_{r,e},\] and denote by \[\pi:=\operatorname{in}(p)\in \operatorname{gr}_\Lambda^1\mathcal{R}_{r,e}\] the initial form of \(p\). Thus \((\operatorname{gr}_\Lambda\mathcal{R}_{r,e})/(\pi)\) is the \(\mathbb{F}_p\)-polynomial ring on the variables \(Y_j\), and throughout the next proposition we record only the \(\pi\)-free unit-coefficient part of the initial form.
Proposition 8 (\(\pi\)-free initial form of the divisible \(B\)-term). Fix \(r\ge1\) and \(e\ge2\). Let \(M\ge1\) be not a power of \(p\), write \[M=M_0+M_1p+\cdots+M_sp^s, \qquad 0\le M_i\le p-1,\] and set \(k=pM\). Assume that \[v_{j+2}(r,e)\le pv_{j+1}(r,e) \qquad (0\le j\le s-1).\] Define \[Y^{[i]}:=\prod_{j\ge0}Y_j^{i_j} \qquad\text{for }i=\sum_{j\ge0} i_jp^j,\] so in particular \(Y^{[0]}=1\), and set \[\mathcal{B}_{M,r,e}(Y):=\frac{(pM)!}{p^e}[y^M] \left(\sum_{i=0}^{M-1}\frac{Y^{[i]}}{(pi)!}y^i\right)^{p^e}.\] Then the degree-\(\Lambda(k)\) class of \(\mathcal{B}_{M,r,e}(Y)\) in \((\operatorname{gr}_\Lambda\mathcal{R}_{r,e})/(\pi)\) is \[-Y^{[M]}.\]
Proof. Expand \(\mathcal{B}_{M,r,e}(Y)\) as a sum over multi-indices \(\eta=(\eta_i)_{0\le i<M}\) with \(\sum_i\eta_i=p^e\) and \(\sum_i i\eta_i=M\). By Lemma 2, every scalar coefficient is \(p\)-integral. If the associated exponent vector in the variables \(Y_j\) is \(u=(u_0,u_1,\ldots)\), then the total sum being \(M\) implies that there is no carry out of the \(p^s\)-place, so only carry levels \(0,\ldots,s-1\) can occur. Lemma 11 therefore gives \[\sum_{j\ge0}u_jv_{j+1}(r,e)\ge \Lambda(k),\] under the displayed hypothesis on the pure-power inequalities. Hence every term has filtered weight at least \(\Lambda(k)\).
Passing to \((\operatorname{gr}_\Lambda\mathcal{R}_{r,e})/(\pi)\) kills every term whose scalar coefficient is divisible by \(p\). If a term survives in degree \(\Lambda(k)\), then its scalar coefficient is a \(p\)-adic unit, so Lemma 13 shows that the underlying addition \(\sum_i i\eta_i=M\) is carry-free. Thus its monomial is already \(Y^{[M]}\).
The sum of the scalar coefficients of all such carry-free terms is the coefficient computed in Lemma 12, namely \(-1\) modulo \(p\). Therefore the degree-\(\Lambda(k)\) class of \(\mathcal{B}_{M,r,e}(Y)\) in \((\operatorname{gr}_\Lambda\mathcal{R}_{r,e})/(\pi)\) is \(-Y^{[M]}\). ◻
Remark 2 (The stable-layer quotient). If \(v_{j+2}(r,e)=pv_{j+1}(r,e)\), then \(Y_j^p\) and \(Y_{j+1}\) have the same filtered degree, so one may pass further to the quotient with relations \(Y_j^p=Y_{j+1}\). By Proposition 8, however, the surviving unit part in degree \(\Lambda(k)\) is already carry-free. Thus this extra quotient is not needed in the scalar computation.
Lemma 14 (Abstract pure-power branch pattern and no tie). Fix integers \(r\ge1\) and \(e\ge2\). Let \((w_n)_{n\ge0}\) be defined by \[w_0=r, \qquad w_n=pw_{n-1}\quad(1\le n<e),\] and, for \(n\ge e\), \[w_n=\min\{\Delta_{n,e}+w_{n-e},\,p w_{n-1}\}.\] Set \[s:=\left\lceil\frac{r}{e}\right\rceil.\] Then the branch word for this abstract recursion is \[(B^{e-1}A)^sB^\infty.\] Equivalently, the \(A\)-branch occurs exactly at \(n=e,2e,\ldots,se\), every other level is \(B\)-dominated, and for every \(n\ge e\) one has \[\Delta_{n,e}+w_{n-e}\ne p w_{n-1}.\] Moreover \[w_{je+t}=p^t\left(r+\frac{p^{j e}-1}{p-1}-e\,j\right) \qquad(0\le j\le s,\;0\le t\le e-1),\] and \(w_n=p^{n-e s}w_{e s}\) for all \(n\ge e s\).
Proof. For \(0\le n<e\), only the \(B\)-branch is present, so \(w_n=p^nr\).
Suppose that \(A\) has already occurred at the levels \(e,2e,\ldots,(j-1)e\). Then \[w_{(j-1)e} =r+\sum_{h=1}^{j-1}(\mu_ep^{(h-1)e}-e) =r+\frac{p^{(j-1)e}-1}{p-1}-e(j-1).\] Since the next \(e-1\) levels after \((j-1)e\) are \(B\)-dominated, we have \(w_{je-1}=p^{e-1}w_{(j-1)e}\). Hence \[\begin{align} \bigl(\Delta_{je,e}+w_{(j-1)e}\bigr)-p w_{je-1} &=\mu_ep^{(j-1)e}-e-(p^e-1)w_{(j-1)e}\\ &=(p^e-1)\bigl(e(j-1)-r\bigr)+\mu_e-e. \end{align}\] Because \(0<\mu_e-e<p^e-1\), this quantity is negative exactly when \(r\ge e(j-1)+1\). Therefore the \(A\)-branch occurs precisely for \(j=1,\ldots,s\), and it never ties with the \(B\)-branch.
If the \(A\)-branch occurs at \(je\), then for \(1\le t\le e-1\) one has \[\begin{align} \bigl(\Delta_{je+t,e}+w_{(j-1)e+t}\bigr)-p w_{je+t-1} &=\Delta_{je+t,e}+p^t w_{(j-1)e}\\ &\qquad -p^t\bigl(\Delta_{je,e}+w_{(j-1)e}\bigr)\\ &=\Delta_{je+t,e}-p^t\Delta_{je,e}\\ &=e(p^t-1)>0, \end{align}\] so the next \(e-1\) levels are \(B\)-dominated. If the comparison at a level \(je\) is \(B\)-dominated, then the same explicit comparison shows that every later multiple of \(e\) is also \(B\)-dominated. The intermediate \(e-1\) levels are again \(B\)-dominated by the displayed formula. This gives the branch word and the explicit formula for \(w_{je+t}\). ◻
Proof of Theorem 3. We verify the induction package \(\mathcal{I}(N)\) by strong induction on \(N\).
The base case \(\mathcal{I}(1)\) follows from 19 : one has \[a_1(r)=-p^r, \qquad \Lambda(1)=v_0=r,\] so (L) holds at \(k=1\). The clauses (D) and (P) are vacuous for \(N=1\), and (S) is tautological for totals \(M\le1\).
Pure powers. Let \((w_n)_{n\ge0}\) be the abstract sequence from Lemma 14. As part of the induction on \(\mathcal{I}(N)\), whenever a pure-power level \(p^j\) has already been treated we also record the equality \(v_j=w_j\).
We treat \(n=1\) first. By Lemma 7 we have \(v_0=w_0=r\) and \(v_1=w_1=pr\). Now fix \(n\ge2\). Assume (L) and (D) hold for every \(k<p^n\), and (P) holds for every pure-power level below \(p^n\). Then (S) is available for all integers whose highest base-\(p\) digit is at most \(p^{n-1}\): all pure-power levels below \(p^n\) have already been treated, so the inequalities \(v_{j+1}\le p v_j\) are known for \(0\le j\le n-2\), and the carry argument gives subadditivity for every sum supported in those digit places.
We begin with the term \(pR_n\) in Proposition 7. Every monomial in \(R_n\) is a product of lower coefficients whose total index is \(p^n\). By the lower-bound induction hypothesis and Lemma 10, its product valuation is at least \(p v_{n-1}\). The extra factor \(p\) in \(pR_n\) raises the valuation to at least \(p v_{n-1}+1\).
For \(p^{r+1}S_n\), every monomial in \(S_n\) is a product of lower coefficients whose total index is \(p^n-1\). Hence (L) for lower indices and (S) at lower levels show that every monomial contribution to \(p^{r+1}S_n\) has valuation at least \[r+1+\Lambda(p^n-1).\] Now \[\Lambda(p^n-1)=(p-1)\sum_{i=0}^{n-1}v_i,\] because the base-\(p\) expansion of \(p^n-1\) has all digits equal to \(p-1\) below level \(n\). Using \(v_{j+1}\le p v_j\) for \(0\le j\le n-2\), we obtain \[\sum_{i=0}^{n-2}(p-1)v_i\ge \sum_{i=0}^{n-2}(v_{i+1}-v_i)=v_{n-1}-v_0.\] Therefore \[\begin{align} r+1+\Lambda(p^n-1) &=1+v_0+(p-1)\sum_{i=0}^{n-1}v_i\\ &\ge 1+v_0+v_{n-1}-v_0+(p-1)v_{n-1}\\ &=1+p v_{n-1}. \end{align}\] Hence every contribution in \(p^{r+1}S_n\) has valuation strictly larger than \(p v_{n-1}\).
If \(1<n<e\), Proposition 7 gives \[a_{p^n}(r)=-\gamma_{n,e}a_{p^{n-1}}(r)^p+pR_n+p^{r+1}S_n.\] The two remainder estimates above show that \[a_{p^n}(r)=-\gamma_{n,e}a_{p^{n-1}}(r)^p+O\bigl(p^{p v_{n-1}+1}\bigr),\] so \[v_n=p v_{n-1}=p w_{n-1}=w_n.\] This gives the case \(n<e\) in (P).
Now assume \(n\ge e\). Substituting the same remainder bounds into Proposition 7, we obtain \[a_{p^n}(r)=\alpha_{n,e}a_{p^{n-e}}(r)-\gamma_{n,e}a_{p^{n-1}}(r)^p+O\bigl(p^{p v_{n-1}+1}\bigr).\] By the already-treated pure-power levels, we have \(v_{n-e}=w_{n-e}\) and \(v_{n-1}=w_{n-1}\). Therefore the two main terms have valuations \[A_{n,e}=\Delta_{n,e}+v_{n-e}=\Delta_{n,e}+w_{n-e}, \qquad B_{n,e}=p v_{n-1}=p w_{n-1}.\] By Lemma 14, these two valuations are never equal. The no-tie assertion is essential here: because the two candidate leading terms have distinct \(p\)-adic valuations, no cancellation can raise the valuation of \(a_{p^n}(r,e)\). Since both remainder terms have valuation strictly larger than \(B_{n,e}\), the dominant term is exactly the smaller of the \(A\)- and \(B\)-branches. Therefore \[v_n=\min\{A_{n,e},B_{n,e}\}=w_n.\] If \(A_{n,e}<B_{n,e}\), the leading term comes from the \(A\)-branch and \[a_{p^n}(r)=\alpha_{n,e}a_{p^{n-e}}(r)+O\bigl(p^{v_n+1}\bigr).\] If \(B_{n,e}<A_{n,e}\), the leading term comes from the \(B\)-branch and \[a_{p^n}(r)=-\gamma_{n,e}a_{p^{n-1}}(r)^p+O\bigl(p^{v_n+1}\bigr).\] Thus the pure-power clause (P) is proved at level \(p^n\), and in all cases \(v_n\le p v_{n-1}\).
Once these inequalities hold up to level \(n\), the carry argument gives (S) for all integers whose base-\(p\) expansion uses only the digits \(1,p,\ldots,p^n\): replacing \(p\) copies of \(p^j\) by one copy of \(p^{j+1}\) can only decrease or preserve the total weight because \(v_{j+1}\le p v_j\).
Divisible non-pure indices. Let \(k=pM\) with \(M\ge1\) and \(k\) not a power of \(p\). Write \[M=M_0+M_1p+\cdots+M_sp^s, \qquad 0\le M_i\le p-1.\] Assume (L) and (D) hold for every \(n'<k\), and (P) holds for every pure-power level below \(k\). Then (S) is available for all integers \(\le k\): all pure-power levels \(p^j\le k\) have already been treated, so the inequalities \(v_{j+1}\le p v_j\) are known up to the relevant level, and the carry argument therefore gives subadditivity for every sum with total at most \(k\).
We have the following shift inequality. If \(M'\ge1\) and \(p^eM'\le k\), then \[\label{eq:shift-ineq-pe} \Lambda(p^eM')\le \mu_eM'-e+\Lambda(M'),\tag{21}\] and the inequality is strict if \(M'\) is not a power of \(p\). Indeed, writing \(M'=\sum_i M'_ip^i\) and using the already-proved pure-power recurrence gives \[\begin{align} \Lambda(p^eM') &=\sum_i M_i'v_{i+e}\\ &\le \sum_i M_i'\bigl(\mu_ep^i-e+v_i\bigr)\\ &=\mu_eM'-e\sum_i M_i'+\Lambda(M'). \end{align}\] Since \(M'\ge1\), one has \(\sum_i M_i'\ge1\), which proves 21 . If \(M'\) is not a power of \(p\), then \(\sum_i M_i'\ge2\), so the inequality is strict.
If \(q\nmid k\), then \(A_k[x^k]=0\). If \(k=qM'\), then \(M'\) is not a power of \(p\), so 21 is strict. Hence \[\mathop{\mathrm{ord}}_p\bigl(A_k[x^k]\bigr)\ge \mu_eM'-e+\Lambda(M')>\Lambda(k).\] For the \(C\)-term, every monomial coefficient is divisible by \(p\) because \(p\mid k\); see Proposition 3. Therefore \[\mathop{\mathrm{ord}}_p\bigl(p^rC_k[x^k]\bigr)\ge 1+r+\Lambda(k-1)>\Lambda(k)\] by (S), since \(k=(k-1)+1\) and \(\Lambda(1)=r\). Thus \[\label{eq:D-first-reduction-pe} a_k(r)\equiv -B_k[x^k]\pmod{p^{\Lambda(k)+1}}.\tag{22}\]
If a monomial in \(B_k[x^k]\) contains some positive index not divisible by \(p\), then, because \(k\ne p\), Proposition 2 implies that its coefficient is divisible by \(p\). The induction hypothesis (L) and (S) give product valuation at least \(\Lambda(k)\), and the coefficient contributes one extra factor of \(p\). Hence modulo \(p^{\Lambda(k)+1}\) one may keep only the divisible truncation and write \[\label{eq:divisible-truncation-pe} B_k[x^k]\equiv \frac{k!}{p^e}[y^M]\left(\sum_{i=0}^{M-1}\frac{a_{pi}(r)}{(pi)!}y^i\right)^{p^e} \pmod{p^{\Lambda(k)+1}}.\tag{23}\] Here the index \(i=0\) is harmless: both sides are \(a_0(r)=m(0)=1\). For \(1\le i<M\), either \(pi\) is a pure power, in which case \(a_{pi}(r)=m(pi)\) by definition, or \(pi\) is divisible and non-pure, in which case the induction hypothesis gives \[a_{pi}(r)\equiv m(pi)\pmod{p^{\Lambda(pi)+1}}.\] Consider a monomial in the expansion of 23 in which at least one factor \(a_{pi}(r)\) is replaced by its error term of valuation at least \(\Lambda(pi)+1\). Every remaining factor has valuation at least its digit weight \(\Lambda(pj)\), and the scalar coefficient is \(p\)-integral by Lemma 2. If the corresponding multiplicities are \(\eta_i\), then subadditivity gives \[1+\sum_i \eta_i\Lambda(pi)\ge 1+\Lambda\!\left(\sum_i \eta_i pi\right)=1+\Lambda(k),\] so the whole contribution is \(O\bigl(p^{\Lambda(k)+1}\bigr)\). Therefore \[\label{eq:replace-by-mr-pe} B_k[x^k]\equiv \frac{k!}{p^e}[y^M]\left(\sum_{i=0}^{M-1}\frac{m(pi)}{(pi)!}y^i\right)^{p^e} \pmod{p^{\Lambda(k)+1}}.\tag{24}\] The already-proved pure-power part of the induction gives \[v_{j+2}\le pv_{j+1} \qquad (0\le j\le s-1),\] so Proposition 8 applies to the polynomial \(\mathcal{B}_{M,r,e}(Y)\). The right-hand side of 24 is obtained from \(\mathcal{B}_{M,r,e}(Y)\) by the filtered specialization \[Y_j\longmapsto a_{p^{j+1}}(r) \qquad (j\ge0),\] for which \(\mathop{\mathrm{ord}}_p(a_{p^{j+1}}(r))=v_{j+1}\) by definition. Hence a term of filtered degree \(>\Lambda(k)\) specializes to \(O\bigl(p^{\Lambda(k)+1}\bigr)\), while the degree-\(\Lambda(k)\) class \(-Y^{[M]}\) specializes to \(-m(k)\). Therefore \[B_k[x^k]\equiv -m(k)\pmod{p^{\Lambda(k)+1}}.\] Combining this with 22 gives \[a_k(r)\equiv m(k)\pmod{p^{\Lambda(k)+1}},\] which proves (D) at the index \(k\).
The global lower bound. Fix \(k\ge1\). Assume (L) holds for every \(n'<k\), (D) holds for every divisible non-pure \(n'<k\), and (P) holds for every pure-power level \(p^j\le k\). Then (S) is available for all integers \(\le k\): all relevant pure-power levels have already been treated, so the inequalities \(v_{j+1}\le p v_j\) are known up to the required height, and the carry argument gives subadditivity for every sum with total at most \(k\).
If \(k=p^n\) is a pure power, then (P) gives \[\mathop{\mathrm{ord}}_p\bigl(a_{p^n}(r)\bigr)=v_n=\Lambda(p^n),\] so the lower bound holds with equality. We may therefore assume that \(k\) is not a pure power. We use the recursion 20 .
If \(q\nmid k\), then \(A_k[x^k]=0\). If \(k=qM\), then by the induction hypothesis and 21 , \[\mathop{\mathrm{ord}}_p\bigl(A_k[x^k]\bigr)\ge \mu_eM-e+\Lambda(M)\ge \Lambda(qM)=\Lambda(k).\]
Take a monomial \(\gamma_B(\eta)a^\eta\) in \(B_k[x^k]\). By Lemma 2, \(\gamma_B(\eta)\in\mathbb{Z}_p\). By the induction hypothesis, \[\mathop{\mathrm{ord}}_p(a^\eta)\ge \sum_{n=0}^{k-1}\eta_n\Lambda(n).\] Since \(\sum_{n=0}^{k-1}\eta_n n=k\), (S) gives \[\sum_{n=0}^{k-1}\eta_n\Lambda(n)\ge \Lambda(k).\] Hence every \(B\)-monomial has valuation at least \(\Lambda(k)\).
Take a monomial \(\gamma_C(\eta)a^\eta\) in \(C_k[x^k]\). By Proposition 3, the coefficient \(\gamma_C(\eta)\) is \(p\)-integral, and by the induction hypothesis, \[\mathop{\mathrm{ord}}_p(a^\eta)\ge \sum_{n=0}^{k-1}\eta_n\Lambda(n)\ge \Lambda(k-1),\] because \(\sum_{n=0}^{k-1}\eta_n n=k-1\). Therefore \[\mathop{\mathrm{ord}}_p\bigl(p^r\gamma_C(\eta)a^\eta\bigr) \ge r+\Lambda(k-1)\ge \Lambda(k),\] where the last inequality is the special case \(k=(k-1)+1\) of (S). Since every summand in 20 has valuation at least \(\Lambda(k)\), so does \(a_k(r)\). This gives (L) and finishes the induction proof of Theorem 3. ◻
Proposition 9 (Pure-power branch pattern). Let \(s=s_{r,e}=\lceil r/e\rceil\). The branch word for the pure-power recursion is \[(B^{e-1}A)^sB^\infty.\] More explicitly, the \(A\)-branch occurs exactly at \(n=e,2e,\ldots,se\); every other layer is \(B\)-dominated. Moreover \[\label{eq:v-explicit-periodic} v_{je+t}=p^t\left(r+\frac{p^{j e}-1}{p-1}-e\,j\right) \qquad(0\le j\le s,\;0\le t\le e-1),\qquad{(7)}\] and \(v_n=p^{n-e s}v_{e s}\) for \(n\ge e s\).
Proof. Let \((w_n)_{n\ge0}\) be the abstract sequence of Lemma 14. By the pure-power clause of Theorem 3, the sequence \(v_n(r,e)\) satisfies the same initial conditions and the same lag-\(e\) recursion as \(w_n\). Hence \(v_n(r,e)=w_n\) for every \(n\ge0\). The explicit branch word and the formula ?? are therefore exactly those of Lemma 14. ◻
Proposition 10 (Normalized pure-power units). For every \(n\ge0\), \[p^{-v_n(r,e)}a_{p^n}(r,e)\equiv \varepsilon_{r,e}(n)\pmod p, \qquad \varepsilon_{r,e}(n)=(-1)^{1+eN_{r,e}(n)}.\]
Proof. The base case \(n=0\) is 19 . The case \(n=1\) is Lemma 7. Along a \(B\)-branch, \[a_{p^n}(r)\equiv-\gamma_{n,e}a_{p^{n-1}}(r)^p \pmod{p^{v_n+1}},\] and \(-\gamma_{n,e}\equiv1\pmod p\), so the normalized unit is unchanged. Along an \(A\)-branch, \[a_{p^n}(r)\equiv\alpha_{n,e}a_{p^{n-e}}(r) \pmod{p^{v_n+1}},\] and \(p^{-\Delta_{n,e}}\alpha_{n,e}\equiv(-1)^e\pmod p\), so the normalized unit is multiplied by \((-1)^e\). By Proposition 9, the number of \(A\)-branches up to level \(n\) is \(N_{r,e}(n)\), giving the formula. ◻
Proof of Theorem 4. Part (a) is Proposition 9. Part (b) is Proposition 10. If \(m=\sum_{i\ge0}m_ip^i\), then Theorem 3(D) gives the displayed leading term for \(a_{pm}(r,e)\) when \(m\) is not a power of \(p\), and Proposition 10 gives it when \(m\) is a power of \(p\). Therefore \(\mathop{\mathrm{ord}}_p(a_k(r,e))=\Lambda_{r,e}(k)\) for every divisible \(k\), which is part (c).
By Proposition 9, \(v_n=\lambda_{r,e}p^n\) for all \(n\ge es_{r,e}\). Hence for \(k=\sum k_ip^i\) the difference \[\Lambda_{r,e}(k)-\lambda_{r,e}k\] depends only on the finitely many digits below level \(es_{r,e}\) and is \(O(1)\). This gives the valuation asymptotic in part (c).
For the radius, write \(f_{r,e}(x)=\sum_{N\ge1}b_Nx^N\) with \(b_{k+1}=a_k(r,e)/k!\). Since \(N=k+1\), replacing \(N\) by \(k\) in the limsup defining the radius does not change the limit. By the global lower bound and Legendre’s formula, \[\mathop{\mathrm{ord}}_p\left(\frac{a_k(r,e)}{k!}\right) \ge \Lambda_{r,e}(k)-\frac{k-S_p(k)}{p-1} \ge -\theta_{r,e}k+O(1).\] Thus \(\rho(f_{r,e})\ge p^{-\theta_{r,e}}\). Along \(k=p^n\) with \(n\ge es_{r,e}\), \[\mathop{\mathrm{ord}}_p\left(\frac{a_{p^n}(r,e)}{(p^n)!}\right) =\lambda_{r,e}p^n-\frac{p^n-1}{p-1} =-\theta_{r,e}p^n+\frac{1}{p-1},\] so the opposite inequality holds. Therefore \(\rho(f_{r,e})=p^{-\theta_{r,e}}\). ◻
Lemma 15 (Tail terms have controlled lower index and integral scalar coefficients). Let \(\vartheta_h\in\mathbb{Q}_p\) for all \(h\ge1\), and consider the formal germ \[\widetilde{\varphi}_{r,e}(x)=x^q+qp^r x^{q+1}+q\sum_{h\ge1}\vartheta_h x^{q+1+h}, \qquad q=p^e,\] with inverse Böttcher coordinate \[\widetilde{f}_{r,e}(x)=x\sum_{k\ge0}\frac{\widetilde{a}_k(r,e)}{k!}x^k, \qquad \widetilde{\varphi}_{r,e}\bigl(\widetilde{f}_{r,e}(x)\bigr)=\widetilde{f}_{r,e}(x^q).\] In the coefficient recursion for \(\widetilde{a}_k(r,e)\), the extra contribution coming from the tail term \(q\vartheta_hx^{q+1+h}\) has the form \(\vartheta_hT_{k,h}(\widetilde{a})\), where \(T_{k,h}(\widetilde{a})=0\) for \(k<h+1\), and every monomial of \(T_{k,h}(\widetilde{a})\) is \[k!\binom{q+1+h}{\eta_0,\eta_1,\ldots} \prod_{n\ge0}\frac{\widetilde{a}_n(r,e)^{\eta_n}}{(n!)^{\eta_n}}\] with \[\sum_{n\ge0}\eta_n=q+1+h, \qquad \sum_{n\ge0}n\eta_n=k-(h+1).\] In particular, every scalar coefficient of \(T_{k,h}(\widetilde{a})\) is an integer, hence \(p\)-integral, and every monomial of \(T_{k,h}(\widetilde{a})\) is a product of lower coefficients with total lower index \(k-(h+1)\).
Proof. If \(k<h+1\), then the coefficient of \(x^k\) in \[x^{q+1+h}\left(\sum_{n\ge0}\widetilde{a}_n(r,e)x^n/n!\right)^{q+1+h}\] vanishes. Hence \(T_{k,h}(\widetilde{a})=0\). Assume now that \(k\ge h+1\), and set \(m:=k-(h+1)\). Then \[T_{k,h}(\widetilde{a})=k![x^m]\left(\sum_{n\ge0}\frac{\widetilde{a}_n(r,e)}{n!}x^n\right)^{q+1+h}.\] The multinomial expansion gives the displayed monomials, with \[\sum_{n\ge0}\eta_n=q+1+h, \qquad \sum_{n\ge0}n\eta_n=m.\] Hence the total lower index is \(m=k-(h+1)\). Set \(Q:=q+1+h\). Then the scalar coefficient may be written as \[k!\binom{Q}{\eta_0,\eta_1,\ldots}\prod_{n\ge0}\frac{1}{(n!)^{\eta_n}} =\frac{k!}{m!}\cdot \frac{Q!}{\eta_0!}\cdot \frac{m!}{\prod_{n\ge1}\eta_n!(n!)^{\eta_n}}.\] Here \(k!/m!\in\mathbb{Z}\) and \(Q!/\eta_0!\in\mathbb{Z}\). The last factor counts set partitions of an \(m\)-element set into blocks of positive sizes \(n\ge1\), repeated \(\eta_n\) times, and is therefore an integer. Hence every scalar coefficient of \(T_{k,h}(\widetilde{a})\) is an integer. ◻
Proposition 11 (Small-tail stability in the higher fibers). Let \(r\ge1\) and let \(\vartheta_h\in\mathbb{Q}_p\) for all \(h\ge1\). Consider the formal germ \[\widetilde{\varphi}_{r,e}(x)=x^q+qp^r x^{q+1}+q\sum_{h\ge1}\vartheta_h x^{q+1+h}, \qquad q=p^e,\] where \[\mathop{\mathrm{ord}}_p(\vartheta_h)\ge \Lambda_{r,e}(h+1)+1 \qquad(h\ge1).\] Write \[\widetilde{f}_{r,e}(x)=x\sum_{k\ge0}\frac{\widetilde{a}_k(r,e)}{k!}x^k, \qquad \widetilde{\varphi}_{r,e}\bigl(\widetilde{f}_{r,e}(x)\bigr)=\widetilde{f}_{r,e}(x^q),\] and define \[\widetilde{m}_{r,e}(k):=\prod_{i\ge0}\widetilde{a}_{p^i}(r,e)^{k_i} \qquad\text{for }k=\sum_{i\ge0}k_ip^i.\] Since only finitely many \(h\) contribute at each fixed coefficient degree, the discussion is formal coefficient-wise. Then the conclusions of Theorem 3 remain valid for the perturbed coefficients \(\widetilde{a}_k(r,e)\), with \(\widetilde{m}_{r,e}(k)\) in place of \(m_{r,e}(k)\). More precisely:
\(\mathop{\mathrm{ord}}_p(\widetilde{a}_k(r,e))\ge \Lambda_{r,e}(k)\) for every \(k\ge1\).
If \(p\mid k\) and \(k\) is not a power of \(p\), then \[\widetilde{a}_k(r,e)\equiv \widetilde{m}_{r,e}(k)\pmod{p^{\Lambda_{r,e}(k)+1}}.\]
For every \(n\ge0\), \[\mathop{\mathrm{ord}}_p\bigl(\widetilde{a}_{p^n}(r,e)\bigr)=v_n(r,e), \qquad p^{-v_n(r,e)}\widetilde{a}_{p^n}(r,e)\equiv \varepsilon_{r,e}(n)\pmod p.\] Consequently, the pure-power branch word, the valuation asymptotic, and the radius formula of Propositions 9 and 10, and Theorem 4 remain unchanged.
Proof. By [17], the perturbed coefficients \(\widetilde{a}_k(r,e)\) are \(p\)-integral. Comparing coefficients in the perturbed Böttcher equation gives \[\label{eq:tail-recursion-higher} \widetilde{a}_k(r,e)=A_k[\widetilde{a}]-B_k[\widetilde{a}]-p^rC_k[\widetilde{a}]-\sum_{1\le h<k}\vartheta_hT_{k,h}(\widetilde{a}),\tag{25}\] where \(A_k\), \(B_k\), and \(C_k\) are the same \(A\)-, \(B\)-, and \(C\)-terms as in the clean family.
First we check that the low-index input is unchanged. Write \[\begin{align} \widetilde{g}_r(x)&:=\sum_{j\ge0}\frac{\widetilde{a}_j(r,e)}{j!}x^j,\qquad A(x):=p^r x\widetilde{g}_r(x),\\ B(x)&:=\sum_{h\ge1}\vartheta_hx^{h+1}\widetilde{g}_r(x)^{h+1}. \end{align}\] Then \[\widetilde{g}_r(x^q)=\widetilde{g}_r(x)^q\bigl(1+qA(x)+qB(x)\bigr),\] and therefore \[\log \widetilde{g}_r(x^q)=q\log \widetilde{g}_r(x)+\log\bigl(1+qA(x)+qB(x)\bigr).\] Fix \(1\le n<p\) and argue by induction on \(n\), exactly as in Lemma 6. After dividing the coefficient relation by \(q\), the clean linear term is still \(-p^r[x^{n-1}]\widetilde{g}_r(x)\). It remains to check that every contribution containing at least one tail factor is \(O(p^{nr+1})\).
For the linear tail term, a summand with \(h+1\le n\) contributes \[\vartheta_h[x^{n-(h+1)}]\widetilde{g}_r(x)^{h+1}.\] Because \(h+1<p\), one has \(\Lambda_{r,e}(h+1)=(h+1)r\), so \[\mathop{\mathrm{ord}}_p(\vartheta_h)\ge \Lambda_{r,e}(h+1)+1=(h+1)r+1.\] A monomial contributing to \([x^{n-(h+1)}]\widetilde{g}_r(x)^{h+1}\) is a product of lower coefficients with total lower index \(n-(h+1)\), hence by the low-index induction has valuation at least \(r(n-(h+1))\). Thus every linear tail contribution has valuation at least \[(h+1)r+1+r(n-(h+1))=nr+1.\]
For the nonlinear terms, after dividing by \(q\) we obtain \[\sum_{m\ge2}(-1)^{m-1}\frac{q^{m-1}}{m}(A(x)+B(x))^m.\] The terms involving only \(A(x)\) are exactly the clean nonlinear terms already handled in Lemma 6. Consider instead a term of order \(m\ge2\) containing at least one factor from \(B(x)\) and contributing to the coefficient of \(x^n\). By the low-index induction, every coefficient of \(x^s\) in \(A(x)=p^r x\widetilde{g}_r(x)\) has valuation at least \(rs\). Likewise, for each fixed \(h\ge1\), every coefficient of \(x^s\) in \(\vartheta_hx^{h+1}\widetilde{g}_r(x)^{h+1}\) has valuation at least \(rs+1\): indeed, the tail factor contributes at least \((h+1)r+1\), and the remaining coefficient factors contribute at least \(r(s-(h+1))\). Hence any product of total degree \(n\) containing at least one \(B\)-factor has valuation at least \(nr+1\). Since \(m\le n<p\), the denominator \(m\) is a \(p\)-adic unit, so the prefactor \(q^{m-1}/m\) has nonnegative \(p\)-adic valuation. Therefore every nonlinear tail contribution is also \(O(p^{nr+1})\).
Thus the proof of Lemma 6 is unchanged, so \(\mathop{\mathrm{ord}}_p(\widetilde{a}_n(r,e))\ge nr\) for \(1\le n<p\). At \(k=p\), the same estimate and Lemma 15 show that every tail term is \(O(p^{pr+1})\), so the proof of Lemma 7 gives \[\mathop{\mathrm{ord}}_p\bigl(\widetilde{a}_p(r,e)\bigr)=pr, \qquad p^{-pr}\widetilde{a}_p(r,e)\equiv-1\pmod p.\]
Now assume that \(\mathop{\mathrm{ord}}_p(\widetilde{a}_n(r,e))\ge \Lambda_{r,e}(n)\) is already known for every \(n<k\). By Lemma 15, every monomial of \(T_{k,h}(\widetilde{a})\) is a product of lower coefficients with total lower index \(k-(h+1)\) and has \(p\)-integral scalar coefficient. Repeatedly applying the clean-family subadditivity from Theorem 3(4) therefore gives \[\mathop{\mathrm{ord}}_p\bigl(T_{k,h}(\widetilde{a})\bigr)\ge \Lambda_{r,e}(k-(h+1)).\] Hence \[\mathop{\mathrm{ord}}_p\bigl(\vartheta_hT_{k,h}(\widetilde{a})\bigr) \ge \Lambda_{r,e}(h+1)+1+\Lambda_{r,e}(k-(h+1)) \ge \Lambda_{r,e}(k)+1.\] Thus the whole tail sum in 25 is \(O(p^{\Lambda_{r,e}(k)+1})\); when \(k=p^n\), this is \(O(p^{v_n(r,e)+1})\).
We now run the same induction package as in the proof of Theorem 3. In the pure-power step, the tail sum is absorbed into the existing \(O(p^{v_n+1})\) error term from Proposition 7, so the same no-tie branch comparison applies. In the divisible non-pure step, the tail sum is absorbed into \(O(p^{\Lambda_{r,e}(k)+1})\), so the same initial-form \(B\)-term argument gives the same leading monomial \(\widetilde{m}_{r,e}(k)\). In the global lower-bound step, the tail sum lies strictly above the required lower bound, so the same \(A\)–\(B\)–\(C\) estimate gives the desired valuation bound. Consequently the analogues of Theorem 3(1)–(4) hold for \(\widetilde{a}_k(r,e)\), with \(\widetilde{m}_{r,e}(k)\) in the divisible non-pure clause. The final assertions about the pure-power branch word, normalized units, valuation asymptotic, and radius then follow exactly as in Propositions 9 and 10, and Theorem 4. ◻
Proposition 12 (Small \(p\)-divisible tails in the special fiber). Let \(\vartheta_h\in\mathbb{Q}_p\) for all \(h\ge1\), and consider the formal germ \[\widetilde{\varphi}_{0,e}(x)=x^q+qx^{q+1}+q\sum_{h\ge1}\vartheta_h x^{q+1+h}, \qquad q=p^e,\] with \(\vartheta_h\in p\mathbb{Z}_p\) for all \(h\ge1\). Write \[\widetilde{f}_{0,e}(x)=x\sum_{k\ge0}\frac{\widetilde{a}_k(e)}{k!}x^k, \qquad \widetilde{\varphi}_{0,e}\bigl(\widetilde{f}_{0,e}(x)\bigr)=\widetilde{f}_{0,e}(x^q),\] and let \(a_k=a_k(0,e)\) denote the clean special-fiber coefficients. Then \[\widetilde{a}_k(e)\equiv a_k\pmod p \qquad(k\ge0).\] In particular, the digit-sum law and the residue-class congruences of Theorems 1 and 2 remain valid for \(\widetilde{a}_k(e)\).
Proof. Since \(\widetilde{\varphi}_{0,e}(x)\in x^q+qx^{q+1}\mathbb{Z}_p[\![x]\!]\), [17] gives \(\widetilde{a}_k(e)\in\mathbb{Z}_p\) for every \(k\). Comparing coefficients in the perturbed Böttcher equation gives \[\label{eq:tail-recursion-special} \widetilde{a}_k(e)=A_k[\widetilde{a}]-B_k[\widetilde{a}]-C_k[\widetilde{a}]-\sum_{1\le h<k}\vartheta_hT_{k,h}(\widetilde{a}),\tag{26}\] where \(A_k\), \(B_k\), and \(C_k\) are the special-fiber terms of Sections 2–6. We claim by induction on \(k\) that \(\widetilde{a}_k(e)\equiv a_k\pmod p\). The case \(k=0\) is tautological.
Assume the claim known for all indices \(<k\). The \(A\)-coefficients are \(p\)-integral, all \(B\)-coefficients are \(p\)-integral by Lemma 2, and all \(C\)-coefficients are \(p\)-integral by Proposition 3. Hence the induction hypothesis gives \[A_k[\widetilde{a}]\equiv A_k[a], \qquad B_k[\widetilde{a}]\equiv B_k[a], \qquad C_k[\widetilde{a}]\equiv C_k[a] \pmod p.\] On the other hand, \(\vartheta_h\in p\mathbb{Z}_p\) and Lemma 15 shows that every scalar coefficient of \(T_{k,h}(\widetilde{a})\) is \(p\)-integral, so \[\vartheta_hT_{k,h}(\widetilde{a})\equiv0\pmod p \qquad(1\le h<k).\] Reducing 26 modulo \(p\) therefore gives \[\widetilde{a}_k(e)\equiv A_k[a]-B_k[a]-C_k[a]=a_k\pmod p.\] This completes the induction. The final sentence follows immediately from Theorems 1 and 2. ◻