January 22, 2026
We study the projective geometry of algebraic neural layers, namely families of maps induced by a polynomial activation function, with particular emphasis on the generic Euclidean Distance degree (\(g\mathrm{ED}\)). This invariant is projective in nature and measures the number of optimal approximations of a general point in the ambient space with respect to a general metric. For a fixed architecture (i.e. fixed width and activation polynomial), we prove that the \(g\mathrm{ED}\) is stably polynomial in the dimensions of the input and output spaces. Moreover, we show that this stable polynomial depends only on the degree of the activation function.
Our approach relies on standard intersection theory on the Nash blow-up, which allows us to express the \(g\mathrm{ED}\) as an intersection number over products of Grassmannians. Stable polynomiality is deduced via equivariant localization, while the reduction to the monomial case follows from an explicit Schubert calculus computation on Grassmannians.
The algebraic geometry of neurovarieties has attracted increasing attention in recent years. This is due to the central role played by algebraic and semi-algebraic models in Deep Learning, supported by density and universal approximation results [1], [2], the existence of global invariants describing the expressive power of architectures [3], the relative tractability of the inevitable singularities these varieties exhibit - that are more delicate to work with in a differential-geometric setting [4] - and the appearance of programmatic works such as [5]. A central invariant in Machine Learning is the \(\mathrm{ED}\)-degree, which measures the number of functions within a model class that optimally fit the training data. This invariant depends sensitively on the choice of a metric on the ambient space; however, for a sufficiently general scalar product its value stabilizes. This general value is shown to be a projective invariant and it is given by the sum of the polar classes of the variety, which we call the generic \(\mathrm{ED}\)-degree of the variety (\(g\mathrm{ED}\)) (see [6] for a general treatment of this topic).
In [7] the authors study the projective geometry of polynomial neural networks in great detail, providing an explicit description of the associated varieties. In particular they obtain closed formulas for the \(g\mathrm{ED}\) in the regime where the output dimension varies, using a direct geometric analysis based on explicit parametrizations. The aim of the present paper is to extend their results and study the stable functional properties of the generic \(\mathrm{ED}\)-invariant for the neurovarieties associated to shallow models with general polynomial activation. From a broader perspective, the questions addressed here fit into a general paradigm in which one fixes the combinatorial or structural parameters of a model and studies how algebraic invariants behave as the ambient dimension grows. In particular, the stable polynomiality phenomena proved in this paper are close in spirit to the results obtained for Gaussian models and semidefinite programming via Schubert calculus on varieties of complete quadrics [8], where polynomiality emerges as a structural feature of the intersection-theoretic description.
The main goal of this paper is to understand how the generic Euclidean Distance degree behaves for algebraic neural layers with polynomial activation. We fix a shallow architecture, meaning that the width of the layer and the degree of the activation function are kept fixed, while the dimensions of the input and output spaces are allowed to vary.
Our first main result establishes a stable polynomiality phenomenon for the generic ED-degree. More precisely, Theorem 26 shows that, for fixed width and activation degree, the generic Euclidean Distance degree \(g\mathrm{ED}\) of the associated neurovariety is eventually given by a polynomial function in the dimensions of the input and output spaces.
Our second main result concerns the dependence of the \(g\mathrm{ED}\) on the activation function. In Theorem 30 we prove that, at the level of generic invariants, the \(g\mathrm{ED}\) depends only on the degree of the activation polynomial and not on its specific monomial support. As a consequence, for the purpose of computing the \(g\mathrm{ED}\), one may restrict to monomial activations of the same degree. This implies a drastic simplification for computational purposes: polynomial activations can be replaced by simple monomials without affecting the asymptotic algebraic complexity.
The proofs of Theorems 26 and 30 rely on a uniform intersection-theoretic description of the \(g\mathrm{ED}\). Using the Nash blow-up and standard geometric constructions recalled in Sections 2.3 and 2.4, we express the \(g\mathrm{ED}\) as an intersection number that can be computed as an integral over products of Grassmannians; see in particular Remark 23. The stable polynomiality in Theorem 26 is obtained by applying equivariant localization techniques developed in Section 4.3. The reduction to the monomial case in Theorem 30 is achieved through an explicit Schubert calculus computation on Grassmannians, based on uniform bounds for the Schubert classes contributing to the relevant intersection numbers (Lemmas 27 and 28).
For completeness, additional technical arguments concerning stabilization and polynomiality are collected in Section 6.
In Sections 2 and 3 we review the definition and the main properties of polynomial layers, their associated geometric realizations, the \(\mathrm{ED}\)-degree and the Nash blow-up and such objects. Even if material in these sections is standard or well-known, we include some proofs either for lack of references or to uniformize the techniques involved. Section 4 is the technical heart of the paper: in 4.1 we relate the Chern-Mather classes to the tautological bundles over a product of Grassmannians, which we then use to write \(g\mathrm{ED}\) as an integral in 4.2. Finally in 4.3 we use the natural action of the algebraic torus on the Grassmannian varieties to reduce the integrals to an estimate of Edidin-Graham equivariant localization formula. Stable polynomiality follows from a direct inspection of the resulting expression. In Section 5, working explicitly with Schubert classes on the Grassmannian base, we show that in the stable range only the higher term of the activation actually contributes to the \(g\mathrm{ED}\).
I am grateful to Kathlén Kohn and Jan Draisma for bringing important references to my attention and for their helpful comments about the first draft of this paper.
We introduce here the central notions of the paper. Further details in the case of monomial activations can be found in [3], [9], [10]. See also [7] for a comprehensive introduction to polynomial neural networks and numerous examples. We restrict to models with no bias and where all neurons have the same activation. Although most of the material in this section is elementary, we include it for completeness since to our knowledge polynomial activations with arbitrary support are not systematically treated in the existing literature.
Definition 1. An algebraic neural layer is a collection of polynomial functions \[f_{\theta_{1},\theta_{2}}:V\to W\] such that
\(V\) and \(W\) are finite dimensional real vector spaces, called the space of inputs and of outputs respectively;
there exists an integer \(k\ge1\), called the width of the layer and a polynomial \(P\in\mathbb{R}\left[z\right]\), called the activation function of the layer, such that for every pair \(\left(\theta_{1},\theta_{2}\right)\in\mathrm{Hom}\left(V,\mathbb{R}^{k}\right)\times\mathrm{Hom}\left(\mathbb{R}^{k},W\right)\) we have \[f_{\theta_{1},\theta_{2}}\left(v\right)=\left(\theta_{2}\circ\underline{P}\circ\theta_{1}\right)\left(v\right)\label{eq:ShallowDefinitionFunctionsParametrized}\tag{1}\] where \(\underline{P}\) means the function that applies \(P\) componentwise.
We will refer to this set of functions as \(\mathcal{F}\left(V,W\right)_{P,k}\). If \(P\) is a monomial \(P\left(z\right)=z^{r}\) we will denote \(\mathcal{F}\left(V,W\right)_{P,k}=\mathcal{F}\left(V,W\right)_{r,k}\) and call it a monomial neural layer of degree \(r\), while when \(r=1\) the layer will be called linear and we will simply write \(\mathcal{F}\left(V,W\right)_{k}\) instead of \(\mathcal{F}\left(V,W\right)_{1,k}\).
Remark 2. For a polynomial \(P\left(z\right)=a_{d}z^{d}+\dots+a_{1}z+a_{0}\) we define its support to be the set \(S\left(P\right)=\left\{ n\,\mid\,a_{n}\neq0\right\}\). Note that, in view of Definition 1 we have \[\mathcal{F}\left(V,W\right)_{P,k}\subseteq W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\label{eq:ShallowMinimalProjectiveAmbientSpace}\tag{2}\] and this inclusion is nondegenerate: to see this we can suppose that \(W\) is one dimensional. Identify \(\mathrm{Sym}^{t}\left(V^{\ast}\right)\simeq\mathbb{R}\left[\underline{x}\right]_{t}\) with the space of polynomials in \(n=\dim V\) variables \(\underline{x}=x_{1},\dots,x_{n}\), this is spanned by powers \(\ell\left(\underline{x}\right)^{t}\) where \(\ell\in\left(\mathbb{R}^{n}\right)^{\ast}\), this follows from polarization formulas [11] (since we are in characteristic 0) or from the fact that the Veronese embedding is nondegenerate. Let \(t\in S\left(P\right)\), hence \(a_{t}\neq0\). Let \(\lambda_{0},\dots,\lambda_{\deg\left(P\right)}\in\mathbb{R}\) be all different, it follows from non-vanishing of the Vandermonde determinant that there exist \(\mu_{0},\dots,\mu_{\deg\left(P\right)}\in\mathbb{R}\) such that \[\sum_{i=0}^{\deg\left(P\right)}\mu_{i}\lambda_{i}^{s}=\delta_{s,t}\] for every \(s=0,\dots,\deg\left(P\right)\). It is straightforward to see that \[\sum_{i=0}^{\deg\left(P\right)}\frac{\mu_{i}}{a_{t}}P\left(\lambda_{i}\ell\left(\underline{x}\right)\right)=\ell\left(\underline{x}\right)^{t}.\] Putting everything together we get \[\mathrm{Span}\left(\mathcal{F}\left(V,W\right)_{P,k}\right)=W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right).\]
Lemma 3. With setting and notations as before we have
if \(\lambda\neq0\) then \(\lambda\mathcal{F}\left(V,W\right)_{P,k}=\mathcal{F}\left(V,W\right)_{P,k}=\mathcal{F}\left(V,W\right)_{\lambda P,k}\)
if \(P\left(z\right)=Q\left(\alpha z\right)\in\mathbb{R}\left[z\right]\) with \(\alpha\neq0\) then \(\mathcal{F}\left(V,W\right)_{P,k}=\mathcal{F}\left(V,W\right)_{Q,k}\).
Proof. Item \(\left(1\right)\) is just the sequence of equalities \[\lambda\left(\theta_{2}\circ\underline{P}\circ\theta_{1}\right)\left(v\right)=\left(\lambda\theta_{2}\circ\underline{P}\circ\theta_{1}\right)\left(v\right)=\left(\theta_{2}\circ\underline{\lambda P}\circ\theta_{1}\right)\left(v\right).\] To prove \(\left(2\right)\), let \(f\in\mathcal{F}\left(V,W\right)_{P,k}\), by definition \(f\left(v\right)=\left(\theta_{2}\circ\underline{P}\circ\theta_{1}\right)\left(v\right)\) for \[\left(\theta_{1},\theta_{2}\right)\in\mathrm{Hom}\left(V,\mathbb{R}^{k}\right)\times\mathrm{Hom}\left(\mathbb{R}^{k},W\right)\] with \(\theta_{1}\left(v\right)=Av\). Then for every \(v\in V\) we have that \(\underline{Q}\left(\alpha\theta_{1}\left(v\right)\right)=\underline{P}\left(\theta_{1}\left(v\right)\right)\) and hence \[f\left(v\right)=\left(\theta_{2}\circ\underline{Q}\circ\alpha\theta_{1}\right)\left(v\right)\] which shows that \(f\in\mathcal{F}\left(V,W\right)_{Q,k}\). ◻
In view of Lemma 3 it makes sense to consider the projectivization \[\mathbb{P}\left(\mathcal{F}\left(V,W\right)_{P,k}\right)\subseteq\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\] which is nondegenerate in view of Remark 2. In particular \(\mathbb{P}\left(\mathcal{F}\left(V,W\right)_{P,k}\right)\) is naturally embedded and spans a space of dimension \[\dim W\cdot\left(\sum_{s\in S\left(P\right)}\binom{\dim V+s-1}{s}\right)-1.\]
We want here to give a geometric perspective on what we have constructed so far. First suppose \(W\) is one-dimensional and \(k=1\) and consider an element \(f\in\mathcal{F}\left(V,W\right)_{P,k}\): its action on \(\ell\in V^{\ast}\) is given by \[f\left(\ell\right)=\sum_{s\in S\left(P\right)}a_{s}\ell^{s},\] hence it gives a map \[\begin{align}\mathbb{P}\left(V^{\ast}\right) & \to\mathbb{P}\left(\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\\ \left[\ell\right] & \mapsto\left[\sum_{s\in S\left(P\right)}a_{s}\ell^{s}\right] \end{align}\] which is well-defined since \(\sum_{s\in S\left(P\right)}a_{s}\ell^{s}\) cannot be 0 in \(\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\) unless \(\ell=0\). If \(\dim W>1\) then there exists \(w\in W\), \(w\neq0\) (since we obviously need to discard the 0 map) with \(f\left(\ell\right)=\sum_{s\in S\left(P\right)}wa_{s}\ell^{s}\), this corresponds to \[\begin{align} \mathbb{P}\left(V^{\ast}\right) & \to\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\\ \left[\ell\right] & \mapsto\left[w\otimes\sum_{s\in S\left(P\right)}a_{s}\ell^{s}\right]. \end{align}\]
Definition 4. With setting and notations as before we define \[X\left(V,W\right)_{P}=\left\{ \left[w\otimes\sum_{s\in S\left(P\right)}a_{s}\ell^{s}\right]\in\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\,\mid\,\left(\left[\ell\right],\left[w\right]\right)\in\mathbb{P}\left(V^{\ast}\right)\times\mathbb{P}\left(W\right)\right\} .\] We call the neurovariety associated with \(V,W\) with polynomial activation \(P\) and \(k=1\) neurons \[\mathcal{M}\left(V,W\right)_{P}=\overline{X\left(V,W\right)_{P}}.\] If \(k>1\) then \(\mathcal{F}\left(V,W\right)_{P,k}\) is the space of functions that can be written as \[f\left(\underline{\ell}\right)=\sum_{i=1}^{k}\sum_{s\in S\left(P\right)}w_{i}a_{s}\ell_{i}^{s},\] for \(\underline{\ell}=\left(\ell_{1},\dots,\ell_{k}\right)\in\mathrm{Hom}\left(V,\mathbb{R}^{k}\right)\simeq V^{\ast}\otimes\mathbb{R}^{k}\simeq\bigoplus_{i=1}^{k}V^{\ast}\) and \(w_{1},\dots,w_{k}\in W\). Geometrically the Zariski closure of the set \[\left\{ \left[\sum_{i=1}^{k}\sum_{s\in S\left(P\right)}w_{i}\otimes a_{s}\ell_{i}^{s}\right]\in\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\,\mid\,\left(\left[\underline{\ell}\right],\left[\underline{w}\right]\right)\in\mathbb{P}\left(P_{k}\left(V,W\right)\right)\right\}\] is the so called \(k\)-secant variety of \(\mathcal{M}\left(V,W\right)_{P}\) which we will denote as \[\sigma_{k}\mathcal{M}\left(V,W\right)_{P}=\mathcal{M}\left(V,W\right)_{P,k}.\]
Lemma 5. Let \(P,Q\in\mathbb{R}\left[z\right]\) be such that \(S\left(P\right)=S\left(Q\right)\), then there exists a projectivity \[\Phi:\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\to\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(Q\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\] such that \(\Phi\left(\mathcal{M}\left(V,W\right)_{P,k}\right)=\mathcal{M}\left(V,W\right)_{Q,k}\).
Proof. Let \(P\left(z\right)=\sum a_{s}z^{n}\) and \(Q\left(z\right)=\sum b_{n}z^{n}\), then for \(s\in S\left(P\right)=S\left(Q\right)\) we let \[\Phi_{s}:\mathrm{Sym}^{s}\left(V^{\ast}\right) \to\mathrm{Sym}^{s}\left(V^{\ast}\right)\qquadbe given by\qquad\Phi \left(v\right)=\frac{b_{s}}{a_{s}}v.\] They are isomorphisms and assemble diagonally to give a projectivity \[\Phi:\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(P\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\to\mathbb{P}\left(W\otimes\bigoplus_{s\in S\left(Q\right)}\mathrm{Sym}^{s}\left(V^{\ast}\right)\right)\] and clearly \(\Phi\left(\mathcal{M}\left(V,W\right)_{P,k}\right)=\mathcal{M}\left(V,W\right)_{Q,k}\). ◻
Since we will deal mainly with generic invariants it makes sense to define
Definition 6. Given a finite set of natural numbers \(S\) we set \[\deg S=\max\left\{ n\,\vert\,n\in S\right\} \quadand\quad V_{S}=\bigoplus_{s\in S}\mathrm{Sym}^{s}\left(V^{\ast}\right).\]
The content of this section is completely standard, see for instance [6] for a thorough introduction or [12] for an approach based on Morse theory to this same argument.
A fundamental notion in the study of neuroalgebraic varieties is that of isotropic quadric associated with a scalar product: since algebraic objects arise as complexification of their counterparts defined over \(\mathbb{R}\), by scalar product we mean the complex-bilinear extension of a real positive definite inner product. In particular, Hermitian forms do not appear in this context.
Definition 7. Let \(q\) be a nondegenerate scalar product on \(\mathbb{C}^{n+1}\), the isotropic quadric associated with \(q\) is the hypersurface \(Q\subseteq\mathbb{P}^{n}\) defined as the zero locus of the quadratic form associated with \(q\).
For a fixed choice of a basis there is a one-to-one correspondence between smooth quadric hypersurfaces \(Q\subseteq\mathbb{P}^{n}\) with no real points and nondegenerate scalar products on \(\mathbb{C}^{n+1}\) up to multiplication by a scalar. For the rest of this section fix a nondegenerate scalar product \(q\) on \(\mathbb{C}^{N+1}\) with isotropic quadric \(Q\subseteq\mathbb{P}^{N}\). We will refer to \(q\) and \(Q\) interchangeably.
Definition 8. Let \(X\subseteq\mathbb{R}^{N}\) be a reduced and irreducible algebraic variety with a smooth real point, denote with \(X_{\mathbb{C}}\subseteq\mathbb{C}^{N}\) its complexification and let \(u\in\mathbb{C}^{N}\). We say that \(x\in X_{\mathbb{C},{\mathrm{reg}}}\) is \(Q\)-critical with respect to \(u\), or just critical if no ambiguity can arise, if \(x-u\) is orthogonal to \(T_{x}X_{\mathbb{C}}\) with respect to \(Q\).
It is clear that this definition makes sense only for \(u\not\in X_{\mathbb{C}}\). Following [6] and the subsequent discussion let \[\mathcal{E}_{X,Q}=\left\{ \left(x,u\right)\in X_{\mathbb{C},{\mathrm{reg}}}\times\mathbb{C}^{N}\,\big|\,x is critical wrt u\right\}\] then the first projection \(\pi_{1}:\mathcal{E}_{X,Q}\to X_{\mathbb{C},{\mathrm{reg}}}\) identifies the fibre \(\pi_{1}^{-1}\left(x\right)=x+N_{x}X\) with the fibre of the affine normal bundle, hence \(\pi_{1}\) is an affine bundle on \(X_{\mathbb{C},{\mathrm{reg}}}\) of rank \(c\), hence \(\mathcal{E}_{X,Q}\) has dimension \(N\), moreover the second projection \(\pi_{2}:\mathcal{E}_{X,Q}\to\mathbb{C}^{N}\) is then dominant between varieties of the same dimension1: it is generically finite and the fibres have the same cardinality for generic \(u\in\mathbb{C}^{N}\). Therefore it makes sense to define
Definition 9. With setting and notations as before, we define \(\mathrm{ED}_{Q}\left(X\right)\) as the cardinality of the generic fibre of the projection \(\pi_{2}:\mathcal{E}_{X,Q}\to\mathbb{C}^{N}\), called the \(\mathrm{ED}\)-degree of \(X\) with respect to \(Q\). If \(X\subseteq\mathbb{P}\left(V\right)\) is a reduced and irreducible real projective variety we define \(\mathrm{ED}_{Q}\left(X\right)\) as the \(\mathrm{ED}\)-degree of its affine cone \(C\left(X_{\mathbb{C}}\right)\subseteq V\otimes\mathbb{C}\) with respect to \(Q\). The bundle \(\pi=\pi_{1}:\mathcal{E}_{X,Q}\to X_{\mathbb{C},\mathrm{reg}}\) is called the \(\mathrm{ED}\)-correspondence of \(X\) with respect to \(Q\). We will denote the \(\mathrm{ED}\)-degree of the \(k\)-th secant variety of the Segre product \(\mathrm{Seg}\left(\mathbb{P}\left(\mathbb{R}^{m}\right)\times v_{r}\left(\mathbb{R}^{n,\ast}\right)\right)\) with respect to \(Q\) as \[\mathrm{ED}_{Q}\left(m,n,k,r\right).\]
Remark 10. Some remarks are in order:
The notion of criticality of a point depends only on the projective class of the scalar product, therefore it makes sense to refer to the isotropic quadric in the definition even if \(\mathrm{ED}_{Q}\) is computed using the affine cone;
in view of [6] the number \(\mathrm{ED}_{Q}\left(X\right)\) counts the number of complex critical points of the distance function \[d_{u}\left(x\right)=\left\Vert x-u\right\Vert ^{2}:X_{\mathbb{C}}\to\mathbb{C}\] for a general \(u\in\mathbb{C}^{N}\backslash X_{\mathbb{C},{\mathrm{reg}}}\). The number of such real points is usually smaller.
the definition in the case of a projective variety has some complication which will not affect us in this paper: one can show (see [6]) that this definition is well posed in case \(X\) is not contained in the isotropic quadric. This is always the case for us since our varieties are real.
While the variety \(X\) is defined over \(\mathbb{R}\), the bundle \(\mathcal{E}_{X,Q}\) is defined over the complexification. If \(X\subseteq\mathbb{P}\left(V\right)\) is projective then the map \(V\backslash\left\{ 0\right\} \to\mathbb{P}\left(V\right)\) induces a vector bundle over \(X_{\mathbb{C},\mathrm{reg}}\) which by abuse of notations, we still denote with \[\mathcal{E}_{X,Q}\to X_{\mathbb{C},\mathrm{reg}}.\] When the variety \(X\) is smooth then \(\mathcal{E}_{X,Q}\) is the total space of the normal bundle of the embedding.
While the \(\mathrm{ED}\)-degree introduced in Section 2.2 is a metric invariant of the variety, we introduce here a purely projective analogue. We fix a nondegenerate scalar product \(q\) with isotropic quadric \(Q\).
Definition 11. Let \(X\subseteq\mathbb{P}^{N}\) be a reduced and irreducible variety, then the conormal variety of \(X\) is defined as
\[\mathcal{N}_{X}=\overline{\{\left(x,H\right)\in\mathbb{P}^{N}\times\left(\mathbb{P}^{N}\right)^{\ast}\,\big|\,x\in X_{\mathrm{reg}}\,,T_{x}X\subseteq H\}}\] where the closure is taken in \(\mathbb{P}^{N}\times\left(\mathbb{P}^{N}\right)^{\ast}\). It comes with the two projections
\[\pi_{X}:\mathcal{N}_{X}\to\mathbb{P}^{N}\quad\quad\pi_{X^{\ast}}:\mathcal{N}_{X}\to\left(\mathbb{P}^{N}\right)^{\ast}.\]
Remark 12. The nondegenerate scalar product on \(\mathbb{C}^{N+1}\) determines an identification between \(\mathbb{P}^{N}\) and \(\left(\mathbb{P}^{N}\right)^{\ast}\) and it is then possible to see \(X\) and \(X^{\ast}\) inside the same projective space via
\[\mathcal{N}_{X}=\overline{\{\left(x,y\right)\in\mathbb{P}^{N}\times\mathbb{P}^{N}\,\big|\,x\in X_{\mathrm{reg}}\,,y\perp_{q}T_{x}X\}}.\] This is the point of view adopted in [6]. In particular, if we denote with \(\mathbb{P}\left(\mathcal{E}_{X,Q}^{(u)}\right)\) the projectivization of the \(\mathrm{ED}\)-correspondence for a generic \(u\in\mathbb{C}^{N+1}\) we have
\[\mathbb{P}\left(\mathcal{E}_{X,Q}^{(u)}\right)=\{\left(x,[y]\right)\in X_{\mathrm{reg}}\times\mathbb{P}^{N}\,\big|\,y\perp_{q}T_{x}X\,,\,y\in\langle x-u\rangle\}.\] It follows that, if \(\Gamma_{u,Q}\subseteq\mathbb{P}^{N}\times\left(\mathbb{P}^{N}\right)^{\ast}\) is the closure of the graph of \(x\mapsto H_{u}(x)\) where \(H_{u}(x)\) is the hyperplane corresponding to \(x-u\),
\[\mathbb{P}\left(\mathcal{E}_{X,Q}^{(u)}\right)=\mathcal{N}_{X}\cap\Gamma_{u,Q}\] where we used the scalar product to identify \(\mathbb{P}^{N}\simeq\left(\mathbb{P}^{N}\right)^{\ast}\). Note that \(\Gamma_{u,Q}\) is rationally equivalent to the diagonal \(\Delta\subseteq\mathbb{P}^{N}\times\mathbb{P}^{N}\).
It is clear that \(\pi_{X}\) defines a morphism \(\pi_{X}:\mathcal{N}_{X}\to X\) and that the restriction
\[\mathcal{N}_{X_{\mathrm{reg}}}=\pi_{X}^{-1}\left(X_{\mathrm{reg}}\right)\to X_{\mathrm{reg}}\] is the projectivized conormal bundle parametrizing hyperplanes tangent to \(X\). We call the image \(\pi_{X^{\ast}}\left(\mathcal{N}_{X}\right)\subseteq\left(\mathbb{P}^{N}\right)^{\ast}\) the dual variety of \(X\), denoted \(X^{\ast}\). If \(\dim X=d\) then for a smooth point \(x\in X\) the fibre \(\pi_{X}^{-1}(x)\) is the projective space of hyperplanes containing \(T_{x}X\) hence has dimension \(N-d-1\). It follows that \(\mathcal{N}_{X}\) is an irreducible variety of dimension \(N-1\). Its conormal cycle, which by an abuse of notation we still denote with \(\mathcal{N}_{X}\), admits a multidegree decomposition in the Chow ring
\[\left[\mathcal{N}_{X}\right]\in A^\bullet\left(\mathbb{P}^{N}\times\left(\mathbb{P}^{N}\right)^{\ast}\right)\simeq\frac{\mathbb{Z}\left[x,y\right]}{\left(x^{N+1},y^{N+1}\right)}\] where \(x,y\) denote the hyperplane classes of the two factors and it makes sense to give the following definition.
Definition 13. With setting and notations as above define \(\delta_{i}\left(X\right)\in\mathbb{Z}\) as
\[\left[\mathcal{N}_{X}\right]=\sum_{i=0}^{d}\delta_{i}\left(X\right)x^{d-i}y^{N-1-d+i}.\] The numbers \(\delta_{i}\left(X\right)\) are called the polar degrees of \(X\) and we define the generic Euclidean Distance degree of \(X\) as
\[g\mathrm{ED}\left(X\right)=\sum_{i=0}^{d}\delta_{i}\left(X\right).\] We will denote the \(g\mathrm{ED}\)-degree of the \(k\)-th secant variety of the Segre product between \(\mathbb{P}\left(\mathbb{R}^{m}\right)\) and \(v_{r}\left(\mathbb{R}^{m,\ast}\right)\) as \[g\mathrm{ED}\left(m,n,k,r\right).\]
Proposition 14. Let \(X\subset\mathbb{P}^{N}\) be a reduced and irreducible variety of dimension \(d\). Fix a nondegenerate scalar product on \(\mathbb{C}^{N+1}\) with isotropic quadric \(Q\), then \[\mathrm{ED}_{Q}(X)\le g\mathrm{ED}(X).\]
Proof. In \(A^{N}(\mathbb{P}^{N}\times\mathbb{P}^{N})\) the diagonal has class \([\Delta]=\sum_{j=0}^{N}x^{j}y^{N-j}\) hence
\[[\mathcal{N}_{X}]\cdot[\Delta]\cdot x=\left(\sum_{i=0}^{d}\delta_{i}(X)\,x^{d-i}y^{N-1-d+i}\right)\left(\sum_{j=0}^{N}x^{j}y^{N-j}\right)x.\] The only monomials contributing to the coefficient of \(x^{N}y^{N}\) are those for which
\[(d-i)+j+1=N\quad\text{and}\quad(N-1-d+i)+(N-j)=N,\] equivalently \(j=N-1-d+i\). For each \(i=0,\dots,d\) there is exactly one such \(j\), hence
\[\deg\bigl([\mathcal{N}_{X}]\cdot[\Delta]\cdot x\bigr)=\sum_{i=0}^{d}\delta_{i}(X)=g\mathrm{ED}(X).\] In view of Remark 10 we see that then \(g\mathrm{ED}(X)=\deg\bigl([\mathcal{N}_{X}]\cdot[\Gamma_{u}]\cdot x\bigr)\), but the \(\mathrm{ED}\)-degree counts only those points counted by \(\deg\bigl([\mathcal{N}_{X}]\cdot[\Gamma_{u}]\cdot x\bigr)\) that are also affine critical points in \(X_{\mathbb{C},\mathrm{reg}}\), hence
\[\mathrm{ED}_{Q}(X)\le\deg\bigl([\mathcal{N}_{X}]\cdot[\Gamma_{u}]\cdot x\bigr)=\deg\bigl([\mathcal{N}_{X}]\cdot[\Delta]\cdot x\bigr)=g\mathrm{ED}(X).\] ◻
Remark 15. The inequality in Proposition 14 can be strict, see for example [6] and the subsequent examples [6]. Sufficient conditions for the inequality to be an equality are that the isotropic quadric associated with the scalar product has transverse intersection with \(X\) and it is disjoint from \(X_{\mathrm{sing}}\) or, more generally, that \(\mathcal{N}_{X}\) does not intersect the diagonal \(\Delta\subseteq\mathbb{P}^{N}\times\mathbb{P}^{N}\) set theoretically (see [6]).
Although well-known, we include a proof of the following Proposition for completeness, as we could not find a reference in the purely projective setting (see for instance [12] for a different approach).
Proposition 16. Let \(X\subset\mathbb{P}^{N}\) be a reduced and irreducible variety. There exists a nonempty Zariski open subset \(U\) in the space of nondegenerate symmetric forms on \(\mathbb{C}^{N+1}\) such that for any scalar product \(q\in U\), the associated conormal variety \(\mathcal{N}_{X}\) does not intersect the diagonal \(\Delta_{q}\subset\mathbb{P}^{N}\times\mathbb{P}^{N}\) induced by the identification \(\mathbb{P}^{N}\simeq\left(\mathbb{P}^{N}\right)^{\ast}\) via \(q\). Consequently, for a generic scalar product, \(\mathrm{ED}_{Q}\left(X\right)=g\mathrm{ED}\left(X\right)\).
Proof. Let \(V=\mathbb{C}^{N+1}\) and let \(\mathcal{S}=\mathbb{P}\left(\mathrm{Sym}^{2}V^{\ast}\right)\) be the space of symmetric forms defined up to scaling. We consider the incidence variety \(\mathcal{Z}\subset\mathcal{N}_{X}\times\mathcal{S}\) defined as:
\[\mathcal{Z}=\left\{ \left(\xi,\left[q\right]\right)\in\mathcal{N}_{X}\times\mathcal{S}\mid\xi\in\Delta_{q}\right\} ,\] where \(\xi=\left(\left[x\right],\left[y\right]\right)\in\mathcal{N}_{X}\subset\mathbb{P}\left(V\right)\times\mathbb{P}\left(V^{\ast}\right)\). It comes with the two projections
\[\xymatrix{ & \mathcal{Z}\ar[dl]_{\pi_{1}}\ar[dr]^{\pi_{2}}\\ \mathcal{N}_{X} & & \mathcal{S} }\] and we proceed by dimension count: first we use \(\pi_{1}\) to determine \(\dim\mathcal{Z}\) that will be less than \(\dim\mathcal{S}\), hence its image cannot cover all the symmetric forms. By definition of projective space, representative vectors \(x\in V\) and \(y\in V^{\ast}\) are non-zero. The condition \(\xi\in\Delta_{q}\) means that the polarity induced by \(q\) maps \(\left[x\right]\) to \(\left[y\right]\), which in coordinates is equivalent to \(qx\in\langle y\rangle\). Let us consider the first projection \(\pi_{1}:\mathcal{Z}\to\mathcal{N}_{X}\). The fiber over a fixed \(\xi\) is the linear subspace of forms \(q\) satisfying \(qx\in\langle y\rangle\). Consider the evaluation map \(\phi_{x}:\mathrm{Sym}^{2}V^{\ast}\to V^{\ast}\) defined by \(q\mapsto qx\). Since \(x\neq0\), this map is surjective. Indeed, given \(\ell\in V^{\ast}\), we choose a basis where \(x=e_{0}\). Define \(q\) such that its first row (and column) corresponds to the components of \(\ell\) and zero elsewhere yields \(qx=\ell\). The condition \(qx\in\langle y\rangle\) is equivalent to saying that the image of \(q\) under \(\phi_{x}\) lies in the \(1\)-dimensional subspace spanned by \(y\). Consider the quotient map \(\pi:V^{\ast}\to V^{\ast}/\langle y\rangle\). Since \(y\neq0\), this is a projection onto a space of dimension \(N\). The composition \(\psi=\pi\circ\phi_{x}:\mathrm{Sym}^{2}V^{\ast}\to V^{\ast}/\langle y\rangle\) is surjective because both \(\phi_{x}\) and \(\pi\) are surjective. The fiber \(\pi_{1}^{-1}\left(\xi\right)\) corresponds to \(\mathbb{P}\left(\ker\psi\right)\). Since \(\psi\) is surjective onto an \(N\)-dimensional space, the kernel has codimension \(N\) in \(\mathrm{Sym}^{2}V^{\ast}\). Thus, for every \(\xi\in\mathcal{N}_{X}\), the fiber in \(\mathcal{S}\) imposes \(N\) independent linear conditions. We compute the dimension of the total space:
\[\dim\mathcal{Z}=\dim\mathcal{N}_{X}+\dim\pi_{1}^{-1}\left(\xi\right)=\left(N-1\right)+\left(\dim\mathcal{S}-N\right)=\dim\mathcal{S}-1.\] Consider now the second projection \(\pi_{2}:\mathcal{Z}\to\mathcal{S}\). Its image is the locus of forms for which the intersection between \(\mathcal{N}_{X}\) and the induced diagonal is non-empty. Since \(\dim\mathcal{Z}<\dim\mathcal{S}\), the closure \(\mathcal{B}=\overline{\pi_{2}\left(\mathcal{Z}\right)}\) is a proper subvariety of \(\mathcal{S}\). Let \(D\subset\mathcal{S}\) be the hypersurface of degenerate forms (\(\det q=0\)), then the set \(U=\mathcal{S}\setminus\left(\mathcal{B}\cup D\right)\) is non-empty and dense and any \(\left[q\right]\in U\), \(q\) is nondegenerate with \(\mathcal{N}_{X}\cap\Delta_{q}=\emptyset\). In view of Remark 15 we have \(\mathrm{ED}_{Q}\left(X\right)=g\mathrm{ED}\left(X\right)\). ◻
We recall here the main points in [13] in order to explicit the generic \(\mathrm{ED}\) degree in terms of intersection theory on the variety. Let \(X\subset\mathbb{P}^{N}\) be a reduced, irreducible and nondegenerate variety of dimension \(d\). It follows from its definition that the conormal variety \(\mathcal{N}_{X}\) of \(X\) is naturally embedded in \(\mathbb{P}^{N}\times\left(\mathbb{P}^{N}\right)^{\ast}\). Following [13] we can see that 2 \[\delta_{k}\left(X\right)=\deg\left(\left[\mathcal{N}_{X}\right]\cdot\left(h^{\vee}\right)^{k+1}\cdot h^{d-k}\right)\label{eq:delta95k}\tag{3}\] where \(h,h^{\vee}\) denote the pullbacks of the hyperplane classes from \(\mathbb{P}^{N}\) and \(\left(\mathbb{P}^{N}\right)^{\ast}\). We call the classes \(M_{k}\left(X\right)=\left[\mathcal{N}_{X}\right]\cdot\left(h^{\vee}\right)^{k+1}\cdot h^{d-k}\) the \(k\)-th polar classes of \(X\).
Remark 17. If \(X\) is smooth they coincide with the degrees of the classical polar varieties defined via generic linear projections.
Denote with \(p_{X}:\mathrm{Nash}\left(X\right)\to X\) the Nash blow-up of \(X\), namely the closed graph of the Gauss map and let \(\nu:\tilde{T}\to\mathrm{Nash}\left(X\right)\) be the Nash bundle, that is the pullback of the tautological subbundle of \(\mathrm{Nash}\left(X\right)\). We call the class
\[c_{M}\left(X\right)=p_{\ast}\left(c\left(\tilde{T}\right)\cap\left[\mathrm{Nash}\left(X\right)\right]\right)\in A_\bullet\left(X\right)\label{eq:PolarDegreeNash}\tag{4}\] the Chern-Mather class of \(X\) and denote with \(c_{i}^{\mathrm{Ma}}\left(X\right)\in A_{i}\left(X\right)\) its \(i\)-dimensional component. This is intrinsic of \(X\), that is independent on the choice of the embedding (see [14]).
Theorem 18. Let \(X\subseteq\mathbb{P}^{N}\) be a reduced, irreducible and non degenerated \(d\)-dimensional variety. We have \[g\mathrm{ED}\left(X\right)=\sum_{j=0}^{d}(-1)^{d+j}\left(2^{j+1}-1\right)\deg\left(c_{j}^{\mathrm{Ma}}\left(X\right)\cdot H^{j}\right)\] where \(H\) is the hyperplane class in \(\mathbb{P}^{N}\).
Proof. This is [15]. ◻
In view of Lemma 5 we have that for any two polynomials \(P,Q\in\mathbb{R}\left[z\right]\) with \(S\left(P\right)=S\left(Q\right)\) we have \[g\mathrm{ED}\left(\mathcal{M}\left(V,W\right)_{S\left(P\right),k}\right)=g\mathrm{ED}\left(\mathcal{M}\left(V,W\right)_{S\left(Q\right),k}\right)\] therefore we fix a finite set \(S\subseteq\mathbb{Z}^{\ge0}\) and two real vector spaces \(V,W\) with \(\dim V=n\), \(\dim W=m\). Set \(N_{S}=\dim V_{S}\) and \[X_{S}=\mathcal{M}\left(V,W\right)_{S,k}\subseteq\mathbb{P}\left(W\otimes V_{S}\right)\simeq\mathbb{P}^{mN_{S}-1}\] where \(\mathcal{M}\left(V,W\right)_{S,k}=\mathcal{M}\left(V,W\right)_{S\left(P\right),k}\) for any polynomial \(P\) with \(S\left(P\right)=S\). Let \[B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right)\] with tautological subbundles \(\mathcal{U}_{S}\) and \(\mathcal{U}_{k}\) respectively and the corresponding tautological quotients \(\mathcal{Q}_{S}\) and \(\mathcal{Q}_{k}\).
Let \(d_{S}=\dim X_{S}\). The general point \(f\in X_{S}\) corresponds to the projective class of a map of rank \(k\). In particular its image and kernel are well-defined vector spaces, then for such a smooth point write \(K=\ker\left(f\right)\), \(I=\mathrm{Im}\left(f\right)\) and denote with \(\mathbb{C}_{\varepsilon}\) the ring of dual numbers over \(\mathbb{C}\). The following result is standard, we include the proof for lack of a reference.
Lemma 19. Let \(f\in X_{S}\) be a smooth point, then \[T_{f}X_{S}\simeq\left\{ g\in\mathrm{Hom}\left(V_{S},W\right)\,\vert\,g\left(\ker f\right)\subseteq\mathrm{Im}\left(f\right)\right\} .\]
Proof. The tangent space \(T_{f}X_{S}\) is the preimage of \(f\) along \(X_{S}\left(\mathbb{C}_{\varepsilon}\right)\to X_{S}\left(\mathbb{C}\right)\), namely \[T_{f}X_{S}=\left\{ f+\varepsilon g\,\vert\,g:V_{S}\to W,\,\mathrm{rk}\left(f+\varepsilon g\right)\le k\right\} .\] Fix a decomposition \(V_{S}=V_{S}'\oplus\ker\left(f\right)\) and \(W=W'\oplus\mathrm{Im}\left(f\right)\) such that \(f_{\vert V_{S}'}:V_{S}'\to\mathrm{Im}\left(f\right)\) is an isomorphism, in particular we can pick basis such that \[f=\left(\begin{array}{cc} 0 & 0\\ \mathrm{id} & 0 \end{array}\right).\] Let us write with respect to the same decomposition \[g=\left(\begin{array}{cc} g_{11} & g_{12}\\ g_{21} & g_{22} \end{array}\right)\quadhence\quad f+\varepsilon g=\left(\begin{array}{cc} \varepsilon g_{11} & \varepsilon g_{12}\\ \mathrm{id}+\varepsilon g_{21} & \varepsilon g_{22} \end{array}\right)\] therefore the statement is \(g_{12}=0\). The submatrix \(\mathrm{id}+\varepsilon g_{21}\) is invertible (with inverse \(\mathrm{id}-\varepsilon g_{21}\)) hence has rank \(k\), therefore every \(\left(k+1\right)\times\left(k+1\right)\) minor containing it has to vanish and from the fact that \(\varepsilon^{2}=0\) we get \(\varepsilon g_{12}=0\). ◻
Inspired by the standard incidence resolutions of determinantal varieties [16] and Lemma 19, we let \[p:Z_{S}=\mathbb{P}\left(\mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)\to B_{S}\] be the projective bundle associated with \(\mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\).
Remark 20. We point out here, for further reference, that \(B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right)\) and \(p:Z_{S}=\mathbb{P}\left(\mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)\to B_{S}\) depend on the support \(S\) only through the dimension \(N_{S}=\dim V_{S}\) and the rank \(k\), that is they depend only on the abstract vector space structure of \(V_{S}\) and not on its realization as a direct sum of symmetric powers.
Proposition 21. There exists a natural projection \(\pi:Z_{S}\to X_{S}\) that realizes \(Z_{S}\) as the Nash blow-up of \(X_{S}\). Moreover \(\pi\) is a rational resolution of singularities.
Proof. Note that \(Z_{S}\) parametrizes linear maps \(f:V_{S}\to W\) that vanish on \(\left(N_{S}-k\right)\)-dimensional subspace \(K\subseteq V_{S}\) and whose image is contained in a \(k\)-dimensional subspace \(I\subseteq W\), in particular \(\mathrm{rk}\left(f\right)\le k\). It follows that \[Z_{S}=\left\{ \left(\left[f\right],K,I\right)\in X_{S}\times B_{S}\,\vert\,K\subseteq\ker\left(f\right),\mathrm{Im}\left(f\right)\subseteq I\right\},\] hence we have a surjective projection \(\pi:Z_{S}\to X_{S}\). Moreover, being defined as a projective bundle over a smooth base, \(Z_{S}\) is smooth. The map \(\pi\) is birational: in fact the general map inside \(X_{S}\) has rank \(k\) hence it determines its image \(\mathrm{Im}\left(f\right)=I\in\mathrm{Gr}_{k}\left(W\right)\) and its kernel \(K=\ker\left(f\right)\in\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\) uniquely. We see that \(X_{S}\) is normal since as in the above description it can be realized as the determinantal variety \[\left\{ f\in\mathrm{Hom}\left(\bigoplus_{s\in S}\mathrm{Sym}^{s}\left(V^{\ast}\right)\to W\right)\,\vert\,\mathrm{rk}\left(f\right)\le k\right\} .\] In view of [16] determinantal varieties admit rational resolution of singularities and for [16] if a variety admits a rational resolution of singularities then every other resolution of singularities is rational. Define now a morphism \[\begin{align}\Phi:Z_{S} & \to\mathrm{Gr}_{d_{S}+1}\left(\mathrm{Hom}\left(V_{S},W\right)\right)\\ \left(\left[f\right],K,I\right) & \mapsto\left\{ \xi\in\mathrm{Hom}\left(V_{S},W\right)\,\vert\,\xi\left(K\right)\subseteq I\right\} \end{align} .\label{eq:NashMap}\tag{5}\] and set \(Z_{S}^{\mathrm{reg}}=\pi^{-1}\left(X_{S,\mathrm{reg}}\right)\). In view of Lemma 19 when \(\left(\left[f\right],K,I\right)\in Z_{S}^{\mathrm{reg}}\) we have \(\Phi\left(\left[f\right],K,I\right)=T_{f}X_{S}\) hence \(\Phi_{\vert Z_{S}^{\mathrm{reg}}}\) coincides with the Gauss map of \(X_{S}\). It follows that the image of \[\left(\pi,\Phi\right):Z_{S}\to X_{S}\times\mathrm{Gr}_{d_{S}+1}\left(\mathrm{Hom}\left(V_{S},W\right)\right)\] contains the graph of the Gauss map over \(X_{S,\mathrm{reg}}\). By definition the Nash blow-up of \(X_{S}\) is the closure of that graph. Since \(\pi\) is birational and \(Z_{S}\) is normal we conclude that \(Z_{S}\) is the Nash blow-up of \(X_{S}\). ◻
It follows from the description in Proposition 21 that we can see the Nash blow-up \(Z_{S}\) of \(X_{S}\) as an incidence variety
\[\xymatrix{ & Z_{S}=\mathbb{P}\left(\mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)\ar[dl]_{p}\ar[dr]^{\pi}\\ B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right) & & X_{S}=\mathcal{M}\left(V,W\right)_{S,k} }\] Denote also \(Z_{S}^{\mathrm{reg}}=\pi^{-1}\left(X_{S,\mathrm{reg}}\right)\). We have exact sequences \[\begin{align}0\to\mathcal{U}_{S}\to V_{S}\otimes\mathcal{O}\to\mathcal{Q}_{S}\to0\\ 0\to\mathcal{U}_{k}\to W\otimes\mathcal{O}\to\mathcal{Q}_{k}\to0 \end{align}\] We now describe a canonical exact sequence on the Nash blow-up that encodes the projective tangent spaces of \(X_{S}\) and will be used throughout the remainder of the paper. In view of Lemma 19 if \(f:V_{S}\to W\) has rank \(k\), hence its projective class defines a smooth point of \(X_{S}\), then \[T_{f}X_{S}=\mathrm{Hom}\left(\frac{V_{S}}{K},I\right)\oplus\mathrm{Hom}\left(\frac{V_{S}}{K},\frac{W}{I}\right)\oplus\mathrm{Hom}\left(K,I\right).\label{eq:tangentSplitting}\tag{6}\] Consider now the bundle \[\mathcal{E}_{S}=p^{\ast}\Biggl(\left(\mathcal{Q}_{S}^{\ast}\otimes\mathcal{U}_{k}\right)\oplus\left(\mathcal{Q}_{S}^{\ast}\otimes\mathcal{Q}_{W}\right)\oplus\left(\mathcal{U}_{S}^{\ast}\otimes\mathcal{U}_{k}\right)\Biggr)\] over \(Z_{S}\): its fiber over \(\left(\left[f\right],K,I\right)\in Z_{S}^{\mathrm{reg}}\) is \(T_{f}X_{S}\). Let now \(\tilde{T}_{S}\) denote the Nash tangent bundle, namely the pullback of the tautological subbundle of \(\mathrm{Gr}_{d_{S}+1}\left(\mathrm{Hom}\left(V_{S},W\right)\right)\) via the map \(\Phi\) in equation \(\left(\ref{eq:NashMap}\right)\) then the two restrictions \[\mathcal{E}_{S\vert Z_{S}^{\mathrm{reg}}}\simeq\tilde{T}_{S\vert Z_{S}^{\mathrm{reg}}}\] naturally coincide. Since \(Z_{S}\) is smooth, hence normal and \(\mathrm{codim}\left(Z_{S}\setminus Z_{S}^{\mathrm{reg}}\right)\ge2\) this identification extends uniquely to an isomorphism over \(Z_{S}\), therefore \(\mathcal{E}_{S}\simeq\tilde{T}_{S}\). Since passing from the affine tangent space \(T_{f}X_{S}\) to the projective one \(T_{\left[f\right]}X_{S}\) amounts to a quotient by the line \(\left\langle f\right\rangle\), we have the exact sequence \[0\to\mathcal{O}_{Z_{S}}\left(-1\right)\to\mathcal{E}_{S}\to\mathcal{T}_{X_{S}}\to0\label{eq:ExactSequence}\tag{7}\] where \(\mathcal{T}_{X_{S}}\) denotes the vector bundle over \(Z_{S}\) whose fibres are the projective tangent spaces \[\left(\mathcal{T}_{X_{S}}\right)_{\left(\left[f\right],K,I\right)}=T_{\left[f\right]}X_{S},\] in particular if \(\xi=c_{1}\left(\mathcal{O}_{Z_{S}}\left(1\right)\right)\), then \[c\left(\mathcal{T}_{X_{S}}\right)=\frac{p^{\ast}\Biggl(c\left(\mathcal{Q}_{S}^{\ast}\otimes\mathcal{U}_{k}\right)c\left(\mathcal{Q}_{S}^{\ast}\otimes\mathcal{Q}_{W}\right)c\left(\mathcal{U}_{S}^{\ast}\otimes\mathcal{U}_{k}\right)\Biggr)}{1-\xi}.\label{eq:ChernTangent}\tag{8}\]
Remark 22. Note that, in view of \(\left(\ref{eq:ChernTangent}\right)\), the total Chern class \(c\left(\mathcal{T}_{X_{S}}\right)\) on the Nash blow-up \(Z_{S}\) can be written as a polynomial in the Chern classes of the tautological bundles \(\mathcal{U}_{S},\mathcal{Q}_{S}\) on \(\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\) and \(\mathcal{U}_{k},\mathcal{Q}_{k}\) on \(\mathrm{Gr}_{k}\left(W\right)\), together with \(\xi=c_{1}\left(\mathcal{O}_{Z_{S}}(1)\right)\). After pushing forward to \(B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right)\) all the resulting intersection numbers depend on the support \(S\) only through the integer \[N_{S}=\dim V_{S}.\] Hence the Chern-Mather class \(c_{M}\left(X_{S}\right)\), the polar degrees \(\delta_{i}\left(X_{S}\right)\) (in view of \(\left(\ref{eq:PolarDegreeNash}\right)\)), and hence the generic Euclidean Distance degree \(g\mathrm{ED}\left(X_{S}\right)\) via Theorem 18, depend on the support \(S\) only through the number \(N_{S}\) (or, equivalently, through the dimension \(N=mN_{S}-1\) of the ambient projective space).
We now explain how the computation of polar degrees of \(X_{S}\) can be reduced to intersection-theoretic calculations on \(\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right)\). Recall that \[c_{M}(X_{S})=\pi_{\ast}\left(c\left(\mathcal{T}_{S}\right)\cap\left[Z_{S}\right]\right),\] where, in view of Proposition 21, \(Z_{S}=\mathbb{P}\left(\mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)\) is the Nash blow-up of \(X_{S}\). Let \(H\) denote the hyperplane class of \(X_{S}\subset\mathbb{P}\left(\mathrm{Hom}\left(V_{S},W\right)\right)\). Then we have \(\pi^{\ast}H=\xi\), where \(\xi=c_{1}\left(\mathcal{O}_{Z_{S}}\left(1\right)\right)\). By the projection formula, for every \(i,j\) we obtain \[\deg\left(c_{d_S-i}^{\mathrm{Ma}}\left(X_{S}\right)\cdot H^{j}\right)=\int_{X_{S}}c_{d_S -i}^{\mathrm{Ma}}\left(X_{S}\right)\cdot H^{j}=\int_{Z_{S}}c_{i}\left(\widetilde{T}_{S}\right)\xi^{j}.\label{eq:tag6100007B44646100007D}\tag{9}\] Using the exact sequence \(\left(\ref{eq:ExactSequence}\right)\) the class \(c_{i}\left(\widetilde{T}_{S}\right)\) can be written as a polynomial in \(\xi\) with coefficients in the Chow ring of \[B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right),\] involving only the Chern classes of the tautological bundles \(\mathcal{U}_{S},\mathcal{Q}_{S},\mathcal{U}_{k}\) and \(\mathcal{Q}_{k}\), hence all integrals reduce to expressions of the form \[\int_{Z_{S}}\xi^{k}\cdot p^{\ast}(\alpha),\quad\alpha\in A^{\ast}\left(B_{S}\right).\] To evaluate such integrals, we use the projective bundle formula for \(p:Z_{S}=\mathbb{P}\left( \mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)\to B_{S}\). Since \(\mathrm{rk}\left( \mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)=k^{2}\) we have \[p_{\ast}\left(\xi^{k^{2}-1+t}\right)=s_{t}\left( \mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right),\quad t\ge0,\] where \(s_{t}\) denotes the \(t\)-th Segre class. It follows that \[\int_{Z_{S}}\xi^{k^{2}-1+t}\cdot p^{\ast}\left(\alpha\right)=\int_{B_{S}}s_{t}\left( \mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\right)\cdot\alpha.\label{eq:tag6100007B44656100007D}\tag{10}\]
Remark 23. Combining (\(\ref{eq:tag6100007B44646100007D}\)) and (\(\ref{eq:tag6100007B44656100007D}\)), we conclude that all polar degrees of \(X_{S}\) are obtained by integrating polynomials in the Chern classes of the tautological bundles over \(B_{S}\). As a consequence there exists a universal polynomial \[P_{r,k}\in\mathbb{Z}\left[x_{1},x_{2},\ldots;y_{1},y_{2},\ldots;z_{1},\ldots,z_{k}\right],\] depending only on \(r\) and \(k\) such that for any finite support \(S\subseteq\mathbb{Z}_{\ge0}\) with \(\deg S=r\) we have \[g\mathrm{ED}\left(X_{S}\right)=\int_{B_{S}}P_{r,k}\left(c\left(\mathcal{U}_{S}\right),c\left(\mathcal{Q}_{S}\right),c\left(\mathcal{U}_{k}\right)\right).\] The dependence on the support \(S\) enters only through the ranks of the tautological bundles, and hence through the relations in the Chow ring of \(B_{S}\).
We now explain how to evaluate the intersection numbers appearing in the previous section and prove stable polynomiality. Since all integrals are taken over the product of Grassmannians \[B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right),\] we use equivariant localization with respect to the standard action of the algebraic torus. We refer to [17] for details, or [18] for a general introduction. Let \[\mathbb{T}_{S}=\left(\mathbb{C}^{\ast}\right)^{N_{S}}\times\left(\mathbb{C}^{\ast}\right)^{m}\] act diagonally on \(V_{S}\) and \(W\), with characters \[u_{1},\dots,u_{N_{S}}\quad\text{and}\quad w_{1},\dots,w_{m},\] respectively. This induces a natural action on \(B_{S}\), as well as on the tautological bundles \(\mathcal{U}_{S},\mathcal{Q}_{S}\) over \(\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\) and \(\mathcal{U}_{k},\mathcal{Q}_{k}\) over \(\mathrm{Gr}_{k}\left(W\right)\). By functoriality, the classes appearing in the previous section admit canonical equivariant lifts to the equivariant Chow ring \(A_{\mathbb{T}_{S}}^{\ast}\left(B_{S}\right)\), see [18]. We have the following localization formula.
Theorem 24. Let \(F_{\mathbb{T}_{S}}\) be the fraction field of the equivariant Chow ring \(R_{\mathbb{T}_{S}}\). Then for every \(\alpha\in A_{\mathbb{T}_{S}}^{\ast}\left(B_{S}\right)\otimes_{\mathbb{Q}}F_{\mathbb{T}_{S}}\) one has
\[\int_{B_{S}}\alpha=\sum_{p\in B_{S}^{\mathbb{T}_{S}}}\frac{\alpha\vert_{p}}{c_{\mathbb{\mathbb{T}_{S}}}\left(T_{p}B_{S}\right)}.\label{eq:tag6100007B44666100007D}\qquad{(1)}\]
Proof. This is [17]. ◻
Remark 25. We point out that the sum in \((\ref{eq:tag6100007B44666100007D})\) is over the \(\mathbb{T}_{S}\)-fixed points of \(B_S\), hence it is finite and \(c_{\mathbb{T}_{S}}\left(T_{p}B\right)\) denotes the equivariant Euler class of the tangent space at \(p\). Since \(p\) is an isolated fixed point, the tangent space \(T_{p}B_{S}\) decomposes as a direct sum of one-dimensional \(\mathbb{T}_{S}\)-representations and its equivariant Euler class is given by the product of the corresponding weights. The fixed points of the \(\mathbb{T}_{S}\)-action on \(B_{S}\) are indexed by pairs of subsets
\[R\subset\left\{ 1,\dots,N_{S}\right\} ,\quad\left|R\right|=N_{S}-k,\quad T\subset\left\{ 1,\dots,m\right\} ,\quad\left|T\right|=k.\] We denote by \(p_{R,T}\) the corresponding fixed point. At \(p_{R,T}\) the fiber of the tautological subbundle \(\mathcal{U}_{S}\) has equivariant Chern roots \(u_{i}\) for \(i\in R\), while \(\mathcal{Q}_{S}\) has equivariant Chern roots \(u_{a}\) for \(a\notin R\). Similarly, the fiber of \(\mathcal{U}_{k}\) has equivariant Chern roots \(w_{\beta}\) for \(\beta\in T\), while \(\mathcal{Q}_{k}\) has equivariant Chern roots \(w_{\gamma}\) for \(\gamma\notin T\).
In view of the canonical isomorphisms \[T_{p_{R,T}}\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\simeq\mathrm{Hom}\left(\mathcal{U}_{S},\mathcal{Q}_{k}\right),\quad T_{p_{R,T}}\mathrm{Gr}_{k}\left(W\right)\simeq\mathrm{Hom}\left(\mathcal{U}_{k},\mathcal{Q}_{k}\right),\] and Remark \(\ref{rem:identificationEulerClassTangent}\), we have \[c_{\mathbb{T}_{S}}\left(T_{p_{R,T}}B_{S}\right)=\left(\prod_{i\in R}\prod_{a\notin R}\left(u_{a}-u_{i}\right)\right)\left(\prod_{\beta\in T}\prod_{\gamma\notin T}\left(w_{\gamma}-w_{\beta}\right)\right).\label{eq:tag6100007B44676100007D}\tag{11}\]
Theorem 26. Let \(S\subseteq\mathbb{Z}^{\ge0}\) be a finite subset and \(k\ge1\) be an integer, then there exists a polynomial \(P_{S,k}\in\mathbb{Q}\left[x,y\right]\) such that for \(m,n\gg0\) the generic Euclidean distance degree \[g\mathrm{ED}\left(m,n,S,k\right)=P_{S,k}\left(m,n\right).\]
Proof. In view of Theorem 18 we can express \(g\mathrm{ED}\left(m,n,S,k\right)\) as a finite linear combination of intersection numbers of the form \[\deg\left(c_{d_S -i}^{\mathrm{Ma}}\left(X\left(V,W\right)_{S,k}\right)\cdot H^{j}\right),\] where \(H\) denotes the hyperplane class and \(c_\bullet^{\mathrm{Ma}}\) the Chern-Mather classes. As shown in Section \(\ref{sec:sec:Reduction}\), each such intersection number can be written as an integral over the incidence variety \(Z_{S}\) and, via the projective bundle formula, reduced to an integral over the product of Grassmannians
\[B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right),\quad N_{S}=\dim V_{S}=\sum_{r\in S}\binom{n+r-1}{r}.\] More precisely, every term appearing in the formula for \(g\mathrm{ED}\left(m,n,S,k\right)\) is a finite sum of integrals of the form \[\int_{B_{S}}\alpha_{N_{S},m},\] where \(\alpha_{N_{S},m}\) is obtained from universal polynomials in the Chern classes of the tautological bundles on \(B_{S}\) and the Segre classes of \(\mathcal{Q}_{S}^{\vee}\otimes\mathcal{U}_{k}\). We observe that the equivariant class \(\alpha_{N_{S},m}\) is constructed universally from Chern classes of the tautological bundles on \(B_S\) together with pullbacks via \(\pi\colon Z_S \to B_S\) and finite products depending only on \(S\) and \(k\). In particular, the total cohomological degree of \(\alpha_{N_S,m}\) is fixed and equals \(\dim Z_S\), hence it is independent of \(m\) and \(n\). After restriction to a fixed point component \(F \subseteq Z_S^{ \mathbb{T}_S}\), each tautological bundle splits as a direct sum of equivariant line bundles, and the Chern classes appearing in \(\iota_F^{\ast}\alpha_{N_S,m}\) become symmetric polynomials in the corresponding equivariant Chern roots. Although the number of equivariant weights grows with \(m\) and \(n\), the total degree of these symmetric polynomials is uniformly bounded by \(\deg\left(\alpha_{N_{S},m}\right)=\dim Z_{S}\). Moreover, the equivariant Euler class \(c_{\mathbb{\mathbb{T}_{S}}}\left(T_{p}B_{S}\right)\) is a product of linear equivariant weights. Therefore, each localization contribution \[\frac{\iota_{F}^{*}\alpha_{N_{S},m}}{c_{\mathbb{\mathbb{T}_{S}}}\left(T_{p}B_{S}\right)}\] can be expressed as a rational symmetric function whose effective total degree is uniformly bounded, depending only on \(S\) and \(k\). Let \(\widetilde{\alpha}_{N_{S},m}\in A_{\mathbb{T}_{S}}^{\ast}\left(B_{S}\right)\) be an equivariant lift, then localization (?? ) together with (11 ) give \[\int_{B_{S}}\alpha_{N_{S},m}=\sum_{R,T}\frac{\widetilde{\alpha}_{N_{S},m}\vert_{p_{R,T}}}{\left(\prod_{i\in R}\prod_{a\notin R}\left(u_{a}-u_{i}\right)\right)\left(\prod_{\beta\in T}\prod_{\gamma\notin T}\left(w_{\gamma}-w_{\beta}\right)\right)}.\] Since \(\alpha_{N_{S},m}\) is constructed from universal polynomials in the Chern classes of the tautological bundles, its restriction to a fixed point depends only on the corresponding equivariant weights and is symmetric in the variables \(\left\{ u_{a}\right\} _{a\notin R}\) and \(\left\{ w_{\gamma}\right\} _{\gamma\notin T}\). As the left-hand side represents an ordinary (nonequivariant) intersection number, it is independent of the equivariant parameters, therefore, we can specialize to \[u_{i}=i,\quad i=1,\dots,N_{S},\quad w_{j}=j,\quad j=1,\dots,m.\] After this specialization, each summand becomes a finite linear combination of symmetric sums over the sets \(\left\{ 1,\dots,N_{S}\right\} \setminus R\) and \(\left\{ 1,\dots,m\right\} \setminus T\) that is each summand is the evaluation of a polynomial symmetric separately in two sets of variables corresponding to the weights of \(V_{S}\) and \(W\) and of bounded total degree depending only on \(S\) and \(k\). In view of Proposition \(\ref{prop:DoublePolynomiality}\), the sum over all fixed points of such expressions agrees, for \(N_{S},m\gg0\), with a single polynomial function in \(\left(N_{S},m\right)\). Since \(N_{S}=\sum_{r\in S}\binom{n+r-1}{r}\) is a polynomial function of \(n\), it follows that \(g\mathrm{ED}\left(m,n,S,k\right)\) agrees, for \(m,n\gg0\), with a polynomial in the two variables \(\left(m,n\right)\). ◻
We want now to see that the statement in Theorem 26 can be actually strengthened: we see that the polynomial \(P_{S,k}\) does not completely depend on \(S\) but only on \(\deg S\), hence in order to perform \(g\mathrm{ED}\) computations one can always suppose that the activation is monomial.
Let us fix two supports \(S\) and \(S'\) such that \[\deg S=\deg\tilde{S}=r\quadand\quad S=\tilde{S}\cup\left\{ d\right\} ,\] that is they differ only by one monomial of lower degree. It follows that we have a natural inclusion \[V_{\tilde{S}}\to V_{S}=V_{\tilde{S}}\oplus\mathrm{Sym}^{d}\left(V^{\ast}\right)\] that translates to Grassmannians into a componentwise closed embedding \[\iota:B_{\tilde{S}}=\mathrm{Gr}_{N_{\tilde{S}}-k}\left(V_{\tilde{S}}\right)\times\mathrm{Gr}_{k}\left(W\right)\hookrightarrow B_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\times\mathrm{Gr}_{k}\left(W\right)\] which is the identity on the second factor, in particular \[\iota^{\ast}\mathcal{U}_{k}=\mathcal{U}_{k}\otimes\mathcal{O}_{B_{\tilde{S}}}\quad\iota^{\ast}\mathcal{Q}_{S}\simeq\mathcal{Q}_{\tilde{S}}\quad\iota^{\ast}\mathcal{U}_{S}\simeq\mathcal{U}_{\tilde{S}}\oplus\left(\mathrm{Sym}^{d}\left(V^{\ast}\right)\otimes\mathcal{O}_{B_{\tilde{S}}}\right).\label{eq:PullbackTautological}\tag{12}\] We recall the universal polynomial \(P_{r,k}\) from Remark 23 that then satisfies \[\iota^{\ast}F_{S}=\iota^{\ast}P_{r,k}\left(c\left(\mathcal{U}_{S}\right),c\left(\mathcal{Q}_{S}\right),c\left(\mathcal{U}_{k}\right)\right)=P_{r,k}\left(c(\mathcal{U}_{\tilde{S}}),c(\mathcal{Q}_{\tilde{S}}),c\left(\mathcal{U}_{k}\right)\right)+R=F_{\tilde{S}}+R,\label{eq:StabilityUnderExtension}\tag{13}\] where \(R\) is a class whose components have Chow degree at least \(k\cdot\mathrm{rk}\left(\mathrm{Sym}^{d}\left(V^{\ast}\right)\right)\). To control the contribution of these terms in intersection numbers, we need a uniform bound on the Chow degree of \(F_{S}\).
For the Chow ring of Grassmannians and the construction of the Schubert basis see for example [14] and [14].
Lemma 27. Let \(r\ge1\) and \(k\ge1\) be integers, \(P_{r,k}\) be the universal polynomial of Remark 23, and denote \[F_{S}=P_{r,k}\left(c\left(\mathcal{U}_{S}\right),c\left(\mathcal{Q}_{S}\right),c\left(\mathcal{U}_{k}\right)\right)\in A^\bullet\left(B_{S}\right).\]. There exists an integer \(T=T\left(r,k\right)\) depending only on \(r\) and \(k\) such that for any finite \(S\subseteq\mathbb{Z}^{\ge0}\) with \(\max(S)=r\)
if \(F_{S}\) is written, via the Künneth decomposition and the Schubert basis on the first factor, as \[F_{S}=\sum_{\lambda}\sigma_{\lambda}\cdot\beta_{\lambda},\quad\beta_{\lambda}\in A^\bullet\left(\mathrm{Gr}_{k}\left(W\right)\right),\] then \(\beta_{\lambda}=0\) for all partitions \(\lambda\) with \(|\lambda|>T\);
let \(\Theta\in A_\bullet\left(B_{S}\right)\) be any cycle. Suppose that the component of \(\Theta\) on the first factor is supported only on Schubert classes \(\sigma_{\mu}\) corresponding to partitions \(\mu\) such that \(\left|\mu^{c}\right|>T\), where \(\mu^{c}\) denotes the complement of \(\mu\) in the \(k\times\left(N_{S}-k\right)\) rectangle. Then \[\int_{B_{S}}F_{S}\cdot\Theta=0.\]
Proof. Set \(G_{S}=\mathrm{Gr}_{N_{S}-k}\left(V_{S}\right)\) and \(H=\mathrm{Gr}_{k}\left(W\right)\) so that \(B_{S}=G_{S}\times H\). Since both Grassmannians admit cellular decompositions, their Chow rings are torsion-free and we have a Künneth isomorphism \(A^\bullet\left(B_{S}\right)\simeq A^\bullet\left(G_{S}\right)\otimes_{\mathbb{Z}}A^\bullet\left(H\right)\). Under this identification, the Chern classes satisfy \[c_{i}\left(\mathcal{U}_{S}\right),c_{i}\left(\mathcal{Q}_{S}\right)\in A^{i}\left(G_{S}\right)\otimes A^{0}\left(H\right),\quad c_{i}\left(\mathcal{U}_{k}\right)\in A^{0}\left(G_{S}\right)\otimes A^{i}\left(H\right).\] Since \(P_{r,k}\) is a fixed polynomial with finitely many monomials, there exists an integer \(T=T\left(r,k\right)\) equal to the maximal Chow degree on \(G_{S}\) among all monomials appearing in \(P_{r,k}\). We conclude the first statement since then every term in \(F_{S}\) has first-factor Chow degree at most \(T\). For the second statement, write \(\Theta\) in the Schubert basis of \(A_\bullet\left(G_{S}\right)\). By the projection formula, the integral \[\int_{B_{S}}F_{S}\cdot\Theta\] is a linear combination of pairings on \(G_{S}\) of the form \[\int_{G_{S}}\sigma_{\lambda}\cdot\sigma_{\mu},\] where \(\sigma_{\lambda}\) comes from \(F_{S}\) and \(\sigma_{\mu}\) from \(\Theta\). By Poincaré duality on the Grassmannian, such a pairing is non-zero if and only if \(\mu=\lambda^{c}\), equivalently \(\left|\mu^{c}\right|=\left|\lambda\right|\). By the first part of the Lemma, all partitions \(\lambda\) appearing in \(F_{S}\) satisfy \(|\lambda|\le T\). Hence, if \(\Theta\) is supported on partitions \(\mu\) with \(\left|\mu^{c}\right|>T\), no such pairing can occur, and the integral vanishes. ◻
Lemma 28. Fix integers \(k\ge1\) and \(T\ge0\). Let \(N'\ge N\ge k+T\) be integers. Consider the closed embedding \[\begin{align}\iota_{N,N'}:\mathrm{Gr}_{N-k}\left(\mathbb{C}^{N}\right) & \longrightarrow\mathrm{Gr}_{N'-k}\left(\mathbb{C}^{N'}\right)\\ \left[U\right] & \longmapsto\left[U\oplus\mathbb{C}^{N'-N}\right]. \end{align}\] Then the induced pullback on Chow rings satisfies:
\(\iota_{N,N'}^{\ast}\left(\sigma_{i}\right)=\sigma_{i}\) for every \(1\le i\le k\);
\(\iota_{N,N'}^{\ast}\left(\sigma_{\lambda}\right)=\sigma_{\lambda}\) for every \(\lambda\) with \(\left|\lambda\right|\le T\);
\(\iota_{N,N'}^{\ast}:A^{\le T}\left(\mathrm{Gr}_{N'-k}\left(\mathbb{C}^{N'}\right)\right)\longrightarrow A^{\le T}\left(\mathrm{Gr}_{N-k}\left(\mathbb{C}^{N}\right)\right)\) is an isomorphism.
Proof. The proof follows from the functoriality of the tautological quotient bundle and Giambelli’s formula, together with the fact that all partitions of size at most \(T\) fit in the \(k\times\left(N-k\right)\) rectangle when \(N\ge k+T\). ◻
Proposition 29. Fix integers \(r\ge1\) and \(k\ge1\), and let \(T=T\left(r,k\right)\) be as in Lemma 27. Let \(V\) be a complex vector space of dimension \(n\), and for any finite support \(S\subseteq\mathbb{Z}^{\ge0}\) with \(\max\left(S\right)=r\) set \[V_{S}=\bigoplus_{d\in S}\mathrm{Sym}^{d}\left(V^{\ast}\right),\quad N_{S}=\dim\left(V_{S}\right).\] Assume that \[\binom{n+r-1}{r}\ge k+T.\] Then the truncation of \(F_{S}\) to first-factor Chow degrees at most \(T\) only depends on \(S\) through \(r\).
Proof. Since \(r\in S\), the space \(V_{S}\) contains \(\mathrm{Sym}^{r}\left(V^{\ast}\right)\) as a direct summand, hence \[N_{S}\ge\binom{n+r-1}{r}\ge k+T.\] In view of Lemmas 27 and 28, all Schubert classes appearing in \(F_{S}\) are stabilized and independent of \(S\) in degrees at most \(T\). ◻
Theorem 30. Fix integers \(r\ge1\) and \(k\ge1\) and let \(S\subseteq\mathbb{Z}^{\ge0}\) be any finite support with \(\deg\left(S\right)=r\). Then for \(m,n\gg0\) we have \[g\mathrm{ED}\left(\mathcal{M}\left(V,W\right)_{S,k}\right)=g\mathrm{ED}\left(\mathcal{M}\left(V,W\right)_{r,k}\right).\]
A polynomial \(F\in\mathbb{Q}[x_{1},\dots,x_{r}]\) is said to be symmetric if it is invariant under the action of the symmetric group. It is classical that the algebra of symmetric polynomials is generated by the power sums
\[p_{d}=\sum_{i=1}^{r}x_{i}^{d},\quad d\ge1.\] In particular, any symmetric polynomial of degree at most \(D\) can be written as a finite linear combination of monomials in the \(p_{d}\) with total degree at most \(D\). Recall Faulhaber’s polynomial
\[F_{p}\left(n\right)=\sum_{k=1}^{n}k^{p}=\frac{1}{p+1}\sum_{r=0}^{p}\binom{p+1}{r}B_{r}n^{p+1-r}\label{eq:Faulhaber}\tag{14}\] where \(B_{r}\) are the Bernoulli numbers. In particular the sum \(\sum_{k=1}^{n}k^{p}\) is a polynomial in \(n\) of degree \(p+1\). More generally, for any fixed integer \(k\ge0\) and any subset \(S\subset\left\{ 1,\dots,N\right\}\) with \(\left|S\right|=k\), we have \[\sum_{i\notin S}i^{a}=F_{a}\left(N\right)-\sum_{i\in S}i^{a}\] The first term is polynomial in \(N\), while the second term is a finite sum of \(k\) monomials of degree \(a\). We now consider functions symmetric separately in two sets of variables.
Proposition 31. Fix integers \(k,D\ge0\). Let \(F\in\mathbb{Q}\left[x_{1},\dots,x_{k},y_{1},\dots,y_{k}\right]\) be a polynomial which is symmetric separately in the two sets of variables and of total degree at most \(D\). For integers \(N,m\) define \[G\left(N,m\right):=\sum_{\substack{A\subset\left\{ 1,\dots,N\right\} \\ \left|A\right|=k } }\sum_{\substack{B\subset\left\{ 1,\dots,m\right\} \\ \left|B\right|=k } }F\left(A,B\right).\] Then there exists a unique polynomial \(P_{k,D}\left(N,m\right)\in\mathbb{Q}\left[N,m\right]\) such that \[G\left(N,m\right)=P_{k,D}\left(N,m\right)\quad\text{for all }N,m\gg0.\]
Proof. Since \(F\) is symmetric separately in the two sets of variables, it can be written as a finite linear combination of products of power sum polynomials in each block. More precisely, there exist finitely many coefficients \(c_{\lambda,\mu}\in\mathbb{Q}\) such that
\[F\left(x,y\right)=\sum_{\lambda,\mu}c_{\lambda,\mu}p_{\lambda}\left(x\right)p_{\mu}\left(y\right),\] where \(p_{\lambda}=\prod_{i}p_{\lambda_{i}}\) denotes a product of power sums, \(p_{d}\left(x\right)=\sum x_{i}^{d}\), and similarly for \(p_{\mu}\left(y\right)\). Moreover, since \(\deg F\le D\) then \(\left|\lambda\right|+\left|\mu\right|\le D\). By linearity, it is sufficient to prove the statement for \(F\left(x,y\right)=p_{\lambda}\left(x\right)p_{\mu}\left(y\right)\). We get
\[G\left(N,m\right)=\left(\sum_{\substack{A\subset\left\{ 1,\dots,N\right\} \\ \left|A\right|=k } }p_{\lambda}\left(A\right)\right)\left(\sum_{\substack{B\subset\left\{ 1,\dots,m\right\} \\ \left|B\right|=k } }p_{\mu}\left(B\right)\right).\] Therefore it suffices to show that, for any partition \(\lambda\), the quantity
\[S_{\lambda}\left(N\right):=\sum_{\substack{A\subset\left\{ 1,\dots,N\right\} \\ \left|A\right|=k } }p_{\lambda}\left(A\right)\] is a polynomial in \(N\) for \(N\) sufficiently large. Write \(\lambda=\left(\lambda_{1},\dots,\lambda_{r}\right)\). By definition,
\[p_{\lambda}\left(A\right)=\prod_{t=1}^{r}\left(\sum_{a\in A}a^{\lambda_{t}}\right)=\sum_{\left(a_{1},\dots,a_{r}\right)\in A^{r}}a_{1}^{\lambda_{1}}\cdots a_{r}^{\lambda_{r}}.\] Hence
\[S_{\lambda}\left(N\right)=\sum_{\substack{A\subset\left\{ 1,\dots,N\right\} \\ \left|A\right|=k } }\sum_{\left(a_{1},\dots,a_{r}\right)\in A^{r}}a_{1}^{\lambda_{1}}\cdots a_{r}^{\lambda_{r}}.\] Exchanging the order of summation, we obtain \[S_{\lambda}\left(N\right)=\sum_{\left(i_{1},\dots,i_{r}\right)\in\left\{ 1,\dots,N\right\} ^{r}}i_{1}^{\lambda_{1}}\cdots i_{r}^{\lambda_{r}}\cdot\#\left\{ A\subset\left\{ 1,\dots,N\right\} :\left|A\right|=k,\,\left\{ i_{1},\dots,i_{r}\right\} \subset A\right\} .\] If \(s=\left|\left\{ i_{1},\dots,i_{r}\right\} \right|\), the number of subsets \(A\subset\left\{ 1,\dots,N\right\}\) of cardinality \(k\) containing these \(s\) elements is \(\binom{N-s}{k-s}\). Therefore, \[S_{\lambda}\left(N\right)=\sum_{\left(i_{1},\dots,i_{r}\right)\in\left\{ 1,\dots,N\right\} ^{r}}\binom{N-\left|\left\{ i_{1},\dots,i_{r}\right\} \right|}{k-\left|\left\{ i_{1},\dots,i_{r}\right\} \right|}\,i_{1}^{\lambda_{1}}\cdots i_{r}^{\lambda_{r}}.\] The quantity \(\left|\left\{ i_{1},\dots,i_{r}\right\} \right|\) depends only on the pattern of equalities among the indices \(i_{1},\dots,i_{r}\). There are finitely many such patterns. For a fixed pattern with \(s\) distinct indices, the corresponding contribution can be written as a finite linear combination of sums of the form
\[\binom{N-s}{k-s}\sum_{\substack{1\le j_{1},\dots,j_{s}\le N\\ \text{distinct} } }j_{1}^{e_{1}}\cdots j_{s}^{e_{s}},\quad e_{1}+\cdots+e_{s}=\left|\lambda\right|.\] By inclusion-exclusion, each sum over distinct indices is a finite linear combination of products of power sums
\[\sum_{j=1}^{N}j^{d},\] with \(d\le\left|\lambda\right|\). By Faulhaber’s formula, these power sums are polynomials in \(N\). Since \(\binom{N-s}{k-s}\) is also a polynomial in \(N\), each contribution above is polynomial in \(N\) for \(N\) sufficiently large. Hence \(S_{\lambda}\left(N\right)\) is a polynomial in \(N\). Applying the same argument to the second factor, we conclude that \(G\left(N,m\right)\) agrees with a polynomial in the pair \(\left(N,m\right)\) for \(N,m\gg0\). The uniqueness of the polynomial follows from the fact that a polynomial in two variables which vanishes on a set of the form \(\left\{ N\ge N_{0},\;m\ge m_{0}\right\}\) must be identically zero. ◻