Proof. Write \(C=C_1\cup\cdots\cup C_a\), where the integral components have degrees \(b_i\) and normalization genera \(g_i\). The Euler characteristics of the normalization and the curve are \(a-\sum_i g_i\) and \(1-p_a(C)\). The normalization exact sequence therefore gives \[\delta(C)=p_a(C)-\sum_{i=1}^a g_i+a-1, \qquad p_a(C)=\frac{(k-1)(k-2)}2.\] In particular, the term \(a-1\) accounts for the disconnected normalization, and the identity includes the contribution of intersections between different components.

For fixed degrees and genera, multiplying the equations of the components defines a morphism from the product of their integral curve loci to the degree-\(k\) linear system. Restricting to distinct factors gives precisely the reduced curves of this type. The image is constructible, and its dimension is at most \[\begin{align*} \sum_{i=1}^a(3b_i+g_i-1) &=3k+\sum_{i=1}^a g_i-a\\ &=\frac{k(k+3)}2-\delta(C). \end{align*}\] There are only finitely many degree and genus types: \(a\leq k\), \(\sum b_i=k\), and \(0\leq g_i\leq(b_i-1)(b_i-2)/2\). The normalization identity determines \(\delta(C)\) from each type. The required locus is therefore a finite union of constructible images with the stated dimension bound. ◻

Proof. The projection \(\pi:C\to\mathbb P^1_{t:u}\) and the bundle \[\mathcal E=\pi_*\mathcal O_C(k-1) =\bigoplus_{j=0}^{k-1}\mathcal O(k-1-j)\] are fixed throughout \(\mathcal F_C\). The entries of \(R_X\) give a map \(\rho_X:\mathcal O^k\to\mathcal E\) on \(\mathbb P^1\). At \(b_0=[1:0]\), trivialize by \(t\) and put \(y=s/t\). For every \(X\), the map on this fiber sends the \(i\)-th standard basis vector to \[ h_i(y)=\prod_{j\ne i}(y+\theta_j) \quad\text{in}\quad K[y]\big/\prod_j(y+\theta_j). \tag{32}\] Evaluation at the distinct roots \(-\theta_i\) shows that these \(k\) classes form a basis. In particular, all the maps \(\rho_X\) restrict to the same isomorphism at \(b_0\). Each \(\rho_X\) is therefore generically full rank and injective as a sheaf map. Its cokernel is torsion of length \[ \mathop{\mathrm{length}}(\mathop{\mathrm{coker}}\rho_X) =\deg\mathcal E =\sum_{j=0}^{k-1}(k-1-j)=N. \tag{33}\]

On \(C\), the adjugate identity gives \[sR_X=-R_X(tD+uX).\] On either affine base chart, the span of the row entries over the base ring is consequently stable under the affine \(s\)-coordinate. This coordinate generates the finite curve algebra over the base, so the span is already the ideal generated over the curve algebra, with its twist. Hence \[\operatorname{im}\rho_X=\pi_*(J'_X(k-1)).\] Finite pushforward is exact, so \[\mathop{\mathrm{coker}}\rho_X=\pi_*(\mathcal O_{Z'_X}(k-1)).\] This pushforward has finite support, and \(\pi\) is finite, so \(\mathcal O_{Z'_X}\) has finite support as well. Twisting preserves length, as does finite pushforward over the algebraically closed ground field. Equation (33) thus proves that \(Z'_X\) is finite of length \(N\).

Suppose now that \(Z'_X=Z'_{X'}\). Their ideal sheaves on the fixed curve are equal, so \(\rho_X\) and \(\rho_{X'}\) have the same image inside the same bundle \(\mathcal E\). Since both maps are injective, there is an automorphism \(U\) of \(\mathcal O^k\) with \[\rho_{X'}=\rho_XU.\] Every such automorphism is an invertible constant matrix. Restricting this equality to \(b_0\), where both maps are the same isomorphism (32), gives \(U=I\) and hence \(\rho_{X'}=\rho_X\).

Multiplication by \(s\) maps \(\mathcal E\) to \(\mathcal E(1)\), and the row identity says \[s\rho_X=-\rho_X(1)(tD+uX).\] The analogous equation for \(X'\) now gives \(\rho_X(1)u(X-X')=0\). The map \(\rho_X(1)\) is injective, so \(u(X-X')=0\) as a matrix of sections of \(\mathcal O(1)\). Since \(u\) is a nonzero section and \(X-X'\) is constant, this forces \(X=X'\). The same argument works after extending the ground field, so the uniqueness statement also holds for geometric points. ◻

Proof. Fix a point of the affine plane and an integer \(\ell\geq1\), and translate the point to the origin. Inside \(\operatorname{Hilb}^{\ell}(\mathbb A^2_K)\), let \(P_\ell\) be the closed locus of quotients on which \((x,y)^\ell\) acts by zero. Its geometric points are precisely the punctual subschemes supported at the origin: in a length-\(\ell\) local algebra, the \(\ell\)-th power of the maximal ideal is zero.

The frame bundle of the tautological quotient algebra over \(P_\ell\) has dimension \(\dim P_\ell+\ell^2\). In a frame, multiplication by the two coordinates gives commuting nilpotent matrices \(A,B\) of size \(\ell\), and the unit gives a cyclic vector \(v\in K^\ell\), meaning \(K[A,B]v=K^\ell\). These data recover the ideal as \[\{f\in K[x,y]:f(A,B)v=0\},\] and recover its frame. Thus the resulting morphism from the frame bundle to the finite-type scheme of such triples is injective on geometric points. Its geometric fibers have dimension zero, which suffices for a dimension upper bound.

There are finitely many nilpotent Jordan types for \(A\). If its linear centralizer has dimension \(j\), then its group centralizer is the open set of invertible elements in that vector space. The conjugacy orbit of \(A\) therefore has dimension \(\ell^2-j\). Within its centralizer, the nilpotent choices of \(B\) have dimension at most \(j-1\): they satisfy \(\det B=0\), whereas the determinant is one at the identity matrix in the same centralizer. This argument works in every characteristic. Consequently the commuting nilpotent pairs have dimension at most \(\ell^2-1\). The choice of the cyclic vector adds at most \(\ell\) parameters, so \[\dim P_\ell+\ell^2\leq\ell^2+\ell-1, \qquad \dim P_\ell\leq\ell-1.\]

Fix \(p\in C\) and choose an affine plane chart around it, translated so that \(p\) is the origin. Inside the Hilbert scheme of this affine part of \(C\), impose the same condition that the \(\ell\)-th power of the maximal ideal of \(p\) acts by zero. The resulting locus is a closed subscheme of \(P_\ell\), and its geometric points are exactly the length-\(\ell\) subschemes of \(C\) supported at \(p\). Its dimension is therefore at most \(\ell-1\), including when \(p\) is singular. Fix a positive length type \(\ell_1+\cdots+\ell_a=N\). Consider ordered distinct points \((p_1,\ldots,p_a)\) of \(C\) together with subschemes of lengths \(\ell_i\) supported at \(p_i\). This incidence is algebraic: containment in the \(\ell_i\)-th infinitesimal neighborhood of the moving point can be imposed using the \(\ell_i\)-th power of the diagonal ideal on \(C\times C\). The support parameters have dimension at most \(a\), and the fibers have dimension at most \(\sum_i(\ell_i-1)=N-a\). The incidence therefore has dimension at most \(N\). Taking disjoint unions of its subschemes maps it to \(\operatorname{Hilb}^N(C)\). Finitely many length types cover all geometric points of this Hilbert scheme, proving the claim. The case \(N=0\) is immediate. ◻

Proof. Fix a curve \(C\) in the image and its fiber \(\mathcal F_C\). Form \[\mathcal I_C= \{(X,Z')\in\mathcal F_C\times\operatorname{Hilb}^N(C): R_X|_{Z'}=0\}.\] This is a closed incidence. Indeed, restrict the row entries to the universal finite flat subscheme over the Hilbert scheme and push forward their common twist. This produces sections of a vector bundle on the product, whose zero locus is \(\mathcal I_C\).

Every matrix \(X\) occurs in the incidence with its subscheme \(Z'_X\). Conversely, row vanishing on \(Z'\) means \(J'_X\subset I_{Z'}\), so \(Z'\subset Z'_X\). The scheme \(Z'\) has length \(N\) by its membership in the Hilbert scheme, and \(Z'_X\) has length \(N\) by Lemma 15. Therefore they are equal. The same lemma shows that each fiber of \(\mathcal I_C\) over \(\operatorname{Hilb}^N(C)\) has at most one geometric point. Such a finite-type fiber has dimension zero when nonempty, even if it is nonreduced. Projection to \(\mathcal F_C\) is surjective on geometric points, so \[\dim\mathcal F_C \leq\dim\mathcal I_C \leq\dim\operatorname{Hilb}^N(C) \leq N,\] where the last inequality is Lemma 16. ◻

Proof of Proposition 13. By (30), every curve in \(f_D(\Gamma_D)\) has normalization defect at least \(T\). Lemma 14 bounds the dimension of this constructible image by \(k(k+3)/2-T\). Each fiber of the restriction of \(f_D\) to \(\Gamma_D\) is contained in a full matrix fiber and has dimension at most \(N\) by Proposition 17. Consequently \[\dim\Gamma_D \leq\frac{k(k+3)}2-T+\frac{k(k-1)}2 =k^2+k-T,\] as asserted. ◻

Proof. Put \(R=\mathbb Z[\theta_1,\ldots,\theta_a]\) and \(S=\operatorname{Spec}R\). The homogeneous ideal \[I=(\partial_{x_1}F,\ldots,\partial_{x_v}F)\subset R[x_1,\ldots,x_v]\] defines a closed subscheme \[ \mathcal Z=\operatorname{Proj}(R[x]/I)\subset\mathbb P^{v-1}_S. \tag{35}\] Its structural morphism \(\pi:\mathcal Z\to S\) is projective, hence proper. Differentiation commutes with specialization, and the closed projective subscheme in (35) commutes with base change. Thus its geometric fibers are exactly the projective gradient-zero schemes of the specialized polynomials.

The derivatives have positive homogeneous degree \(r-1\). At \(\bar\theta\), their affine zero set is a cone. If its projectivization is nonempty, its dimension is one less than that of the cone: on each projective coordinate chart the corresponding punctured cone is a product with \(\mathbb G_m\). If the projectivization is empty, the same upper bound holds with our convention. Hence \[\dim\mathcal Z_{\bar\theta}\le v-1-\delta.\] Let \(s\in S\) be the image of the geometric point \(\bar\theta\). The dimension of a scheme of finite type over a field is unchanged by extending that field, so the same bound holds for \(\mathcal Z_s\). If \(v-1-\delta\ge0\), upper semicontinuity of fiber dimension for the proper morphism \(\pi\) gives an open neighborhood \(U_S\) of \(s\) on which this bound holds (The Stacks Project Authors 2026, Tag 0D4I). If \(v-1-\delta=-1\), the fiber \(\mathcal Z_s\) is empty. Properness makes \(\pi(\mathcal Z)\) closed, so \(U_S=S\setminus\pi(\mathcal Z)\) is an open neighborhood of \(s\) whose fibers are all empty. Thus the required neighborhood exists in every case, without a flatness assumption.

Since \(S\) is integral, \(U_S\) contains its generic point, which has characteristic zero. To see explicitly that the open has complex points, choose a nonempty principal open \(D(h)\subset U_S\), with \(0\ne h\in R\). The polynomial \(h\) remains nonzero over \(\mathbb C\), so \(D(h)_\mathbb C\) is nonempty. We may take \(U=(U_S)_\mathbb C\). The same invariance under field extension shows that every complex fiber over \(U\) satisfies \[ \dim V_{\mathbb P^{v-1}_{\mathbb C}}(\nabla F_\theta) \le v-1-\delta. \tag{36}\]

Since \(\delta>0\), \(F_\theta\) cannot be the zero polynomial: otherwise the left side of (36) would be \(v-1\). Over \(\mathbb C\), Euler’s identity gives \[rF_\theta=\sum_{i=1}^v x_i\partial_{x_i}F_\theta.\] Thus the ideals \((F_\theta,\nabla F_\theta)\) and \((\nabla F_\theta)\) agree. These equations describe the actual projective hypersurface singular scheme. Indeed, on the chart \(x_i\ne0\), put \(z_j=x_j/x_i\) for \(j\ne i\) and \(f_i=F_\theta/x_i^r\). The hypersurface has equation \(f_i=0\) and cotangent presentation \[\mathcal O_{V(f_i)} \xrightarrow{\,df_i\,} \mathcal O_{V(f_i)}^{\,v-1} \longrightarrow\Omega_{V(f_i)/\mathbb C}\longrightarrow0.\] Its \((v-2)\)-nd Fitting ideal is generated by the local partial derivatives of \(f_i\). On \(f_i=0\), these are the homogeneous gradient equations in this chart: Euler’s identity supplies the derivative in the omitted \(x_i\) direction. The Jacobian criterion therefore identifies this scheme with the singular scheme. Equation (36) proves (34). ◻

Proof. Write \(\mathcal G=\operatorname{Gr}(d,V)\), so \(\dim\mathcal G=d(v-d)\). A point of \(\mathcal G\) is a vector subspace of dimension \(d\), whose projectivization has dimension \(d-1\).

First consider pairs \((p,H)\) with \(p\in\operatorname{Sing}Y\) and \(p\in\mathbb P(H)\). For fixed \(p\), the choices of \(H\) form \(\operatorname{Gr}(d-1,V/p)\), of dimension \((d-1)(v-d)\); here \(p\) also denotes its line in \(V\). This incidence has dimension at most \[(v-1-\delta)+(d-1)(v-d) =\dim\mathcal G+d-1-\delta <\dim\mathcal G.\] Its projection to \(\mathcal G\) is closed because \(\operatorname{Sing}Y\) is projective. It is a proper subset, so outside it the section avoids every singular point of \(Y\).

For \(p=[x]\in Y_{\rm reg}\), the nonzero differential \(dF_x\) has a kernel \(K_p\subset V\) of dimension \(v-1\). Euler’s identity at \(F(x)=0\) shows that \(p\subset K_p\). If \(p\subset H\), the restricted equation is singular at \(p\) exactly when \[p\subset H\subset K_p.\] The homogeneous partial derivatives define the Gauss morphism \(Y_{\rm reg}\to\mathbb P(V^*)\), \(p=[x]\mapsto[dF_x]\). The pairs above form a closed incidence in \(Y_{\rm reg}\times\mathcal G\): they are the inverse image of the closed flag incidence \(p\subset H\subset\ker\ell\) under \((p,H)\mapsto(p,H,[dF_x])\). For fixed \(p\), these subspaces form \(\operatorname{Gr}(d-1,K_p/p)\), of dimension \((d-1)(v-d-1)\). Since \(\dim Y_{\rm reg}\le v-2\), this tangency incidence has dimension at most \[(v-2)+(d-1)(v-d-1)=d(v-d)-1.\] Its image in \(\mathcal G\) is constructible, and the closure of that image has dimension at most \(\dim\mathcal G-1\). Removing this closure and the preceding singular-point incidence image leaves a nonempty open subset of the irreducible Grassmannian.

For \(H\) in that open, every zero of \(F|_H\) has nonzero differential on \(H\). The restriction cannot vanish identically: if it did, every point \(p\in\mathbb P(H)\) would lie in \(Y\), singular avoidance would put it in \(Y_{\rm reg}\), and differentiation of the zero restriction would force \(H\subset K_p\), contrary to the tangency exclusion.

Thus \(F|_H\) is a nonzero homogeneous polynomial of degree \(r\). Since \(d\ge2\), it defines a nonempty hypersurface of dimension \(d-2\) in \(\mathbb P(H)\). The nonzero differentials prove its smoothness. ◻

Proof. Set \(T=\binom{q+2}{2}\). Since \(k\ge128\), \[q\ge\frac{k-4}{3}\ge\frac{k}{4},\qquad T\ge\frac{q^2}{2}\ge\frac{k^2}{32}.\] Consequently \[ \Delta:=T-k \ge\frac{k^2}{32}-k \ge\frac{k^2}{64}>0. \tag{37}\] Also \(T\le(k+2)(k+5)/18<k^2\) for these \(k\), so \(1\le\Delta\le k^2\).

Choose pairwise distinct \(\bar\theta_1,\ldots,\bar\theta_k\in\overline{\mathbb F}_2\) and put \(\bar D=\operatorname{diag}(\bar\theta_1,\ldots,\bar\theta_k)\). Proposition 13 gives \[\dim V_{\mathbb A^{k^2}_{\overline{\mathbb F}_2}}(\nabla G_{\bar D}) \le k^2+k-T=k^2-\Delta.\] The coefficient formula (3), with the \(\theta_i\) treated as indeterminates, is integral in these parameters and homogeneous of degree \(r\) in the \(k^2\) matrix variables. Moreover, \[r-1=k-2q-1 \ge\frac{k-1}{3}\ge\frac{k}{4},\] so \(r\ge2\). Lemma 18 supplies a nonempty open subset \(U\subset\mathbb A^k_{\mathbb C}\). Choose \(\theta=(\theta_1,\ldots,\theta_k)\in U(\mathbb C)\) and put \(D=\operatorname{diag}(\theta_1,\ldots,\theta_k)\). Then \(G_D\ne0\) and \[\dim\operatorname{Sing}V_{\mathbb P^{k^2-1}_{\mathbb C}}(G_D) \le k^2-1-\Delta.\] The complex coefficient projection permits arbitrary diagonal entries for this \(D\).

For \(d=\lfloor k^2/100\rfloor\), we have \(2\le d<k^2\), and (37) gives \[d-1<\frac{k^2}{100}<\frac{k^2}{64}\le\Delta.\] Proposition 19 therefore supplies a subspace \(H\subset\mathop{\mathrm{Mat}}_k(\mathbb C)\) of dimension \(d\) such that \(G_D|_H\) is nonzero and defines a smooth degree-\(r\) hypersurface in \(\mathbb P(H)\). Choose a linear isomorphism \(\Lambda:\mathbb C^d\to H\) and set \(G=G_D\circ\Lambda\). This gives the asserted smooth form. The remaining scale estimate is \[d-1\ge\frac{k^2}{100}-2\ge\frac{k^2}{200},\] where the last inequality holds for \(k\ge20\).

Finally, \(0\le3q\le k-1\), so \(0\le q\le\lfloor k/2\rfloor\). For each integer \(m\ge11k-2q\), Proposition 5 gives a matrix \(A_m(X)\) of affine-linear forms with \[\mathop{\mathrm{per}}_m(A_m(X))=a_0G_D(X),\qquad a_0\ne0.\] Compose \(A_m\) with \(\Lambda\) and multiply its first row by \(a_0^{-1}\). The resulting affine map \(\Phi_m:\mathbb C^d\to\mathop{\mathrm{Mat}}_m(\mathbb C)\) satisfies \(\mathop{\mathrm{per}}_m\circ\Phi_m=G\), as required. ◻

Proof. The sections \(\partial_wG\) of \(\mathcal O_Y(r-1)\) have no common zero on \(Y\): otherwise all partial derivatives of \(G\) would vanish at a point of the smooth hypersurface. We spell out why general choices give simple points. If \(d=2\), the smooth degree-\(r\) hypersurface in \(\mathbb P^1\) already consists of \(r\) distinct points, and there are no polar equations. Assume \(d\ge3\).

Put \(\mathcal P=(\mathbb C^d)^{d-2}\) and consider \[\mathcal I=\{(x,w_1,\ldots,w_{d-2})\in Y\times\mathcal P: \partial_{w_i}G(x)=0\text{ for every }i\}.\] For each \(x\), the equation in each separate \(w_i\) block is a nonzero linear equation. Thus \(\mathcal I\) is the total space of a vector bundle of rank \((d-1)(d-2)\) over \(Y\). In particular it is smooth, and \(\dim\mathcal I=d(d-2)=\dim\mathcal P\). The incidence is closed in \(Y\times\mathcal P\), so its projection to \(\mathcal P\) is proper because \(Y\) is projective. It is surjective: \(G\) and any \(d-2\) chosen polars are \(d-1\) positive-degree homogeneous equations in \(d\) variables, so the projective dimension theorem gives a common projective zero.

Characteristic-zero generic smoothness makes the generic fiber of each dominating component smooth. Remove the proper images of the non-dominating components and of the locus where the projection is not smooth. Properness makes these images closed, and generic smoothness ensures that they omit the generic point of \(\mathcal P\). The resulting nonempty open subset has smooth zero-dimensional fibers. Each such fiber is a finite set of reduced points. At each point the differentials of the polar sections on \(Y\) are independent. Since \(dG\ne0\) in local projective coordinates, the full system (38) is simple there.

This is a proper intersection of hypersurfaces of degrees \(r,r-1,\ldots,r-1\) in \(\mathbb P^{d-1}\). Projective Bézout therefore gives exactly \(r(r-1)^{d-2}\) points, each of multiplicity one. Finally, the linear forms vanishing at at least one of these finitely many points form a finite union of proper hyperplanes in \((\mathbb C^d)^*\). Choose \(L_0\) outside that union. ◻

Proof. First construct a system with the displayed number of zeros, all simple. Give each occurrence of a coordinate group in an equation a different label. On \(\mathbb P^{a_j}\) choose the labeled linear forms so that every subfamily of at most \(a_j+1\) coefficient vectors is independent. In coordinates \([z_0:\cdots:z_{a_j}]\), forms \[z_0+t z_1+\cdots+t^{a_j}z_{a_j}\] with distinct values of \(t\) have this property by the Vandermonde determinant. When \(a_j=0\), use the nowhere-zero form \(z_0\) on \(\mathbb P^0\). Replace equation \(\ell\) by \(Q_\ell=L_\ell R_\ell\), using its two labeled forms on factors \(u_\ell\) and \(v_\ell\).

At a common zero of all \(Q_\ell\), let \(b_j\) count the labeled forms vanishing in factor \(j\). General position implies \(b_j\le a_j\). Each of the \(M\) equations requires at least one vanishing form, and no label occurs in two equations. Hence \[M\le\sum_j b_j\le\sum_j a_j=M.\] Exactly one labeled factor vanishes in each equation, and exactly \(a_j\) forms vanish in factor \(j\). Conversely, choosing one factor of each equation, with exactly \(a_j\) choices assigned to factor \(j\), determines one point of \(Y\): the selected \(a_j\) hyperplanes in that factor meet in one point. No additional labeled form vanishes there, because \(a_j+1\) such hyperplanes have empty intersection. Different choices therefore give different points. Expanding the product in (39) counts exactly these choices; denote that coefficient by \(B\).

Every comparison zero is simple. In equation \(\ell\), one factor is zero and the other is nonzero, so its differential is a nonzero multiple of the selected linear form’s differential. In each projective factor the selected hyperplanes have independent differentials. The full Jacobian is therefore invertible. This construction also covers \(B=0\), when the comparison system has no common zero.

It remains to compare an arbitrary system with this one. Interpolate by \[F_{\ell,t}=(1-t)Q_\ell+tF_\ell\qquad(t\in\mathbb C).\] The common-zero incidence is closed in \(Y\times\mathbb C\). On it, rank deficiency of the differential in the \(Y\) directions is a closed algebraic condition. Indeed, it is given by maximal Jacobian minors in each projective chart. Coordinate changes act invertibly on the differential, and changing a trivialization multiplies an equation by a unit. At a common zero the derivative of that unit contributes zero, so the rank condition is independent of the chart.

Since \(Y\) is projective, the parameter values having a rank-deficient common zero form a Zariski-closed subset \(\mathcal B\subset\mathbb C\). The comparison system shows that \(0\notin\mathcal B\), so \(\mathcal B\) is finite. For \(t\notin\mathcal B\), all zeros are isolated by the implicit function theorem. Their set is closed in the compact space \(Y\), hence finite. The number of zeros is locally constant on \(\mathbb C\setminus\mathcal B\): the implicit function theorem preserves each zero in a disjoint neighborhood, and compactness prevents additional zeros from appearing outside those neighborhoods as the parameter varies. The same argument covers an empty fiber. Because the complement of a finite subset of \(\mathbb C\) is connected, this number is always the number at \(t=0\), namely \(B\).

Finally, choose any finite set of simple zeros at \(t=1\). They persist in disjoint neighborhoods for parameters sufficiently close to \(1\). Such parameters can be chosen outside \(\mathcal B\), where the total count is \(B\). The selected set has at most \(B\) members. This conclusion does not assign multiplicities to, or impose conditions on, any excess components at \(t=1\). ◻

Proof. Choose one sufficiently close determinant. Fix \(\mu\) distinct simple zeros of (40). All output polynomials lie in the finite-dimensional coefficient space of degree at most \(n\). With \(\Psi\) and the \(w_i\) fixed, the map \[(Q,\xi)\longmapsto \bigl(Q(\Psi(\xi)),(\partial_{w_1}Q)(\Psi(\xi)),\ldots, (\partial_{w_h}Q)(\Psi(\xi))\bigr)\] is polynomial in the coefficients of \(Q\) and the coordinates of \(\xi\). At every selected zero for \(P\), its \(\xi\)-Jacobian is invertible. The complex implicit function theorem preserves that simple zero for all sufficiently close coefficient vectors. Take disjoint neighborhoods of the \(\mu\) zeros and intersect their finitely many coefficient neighborhoods. There is an actual determinant in the resulting neighborhood, \[ Q(x)=\det A(x),\qquad A(x)=A_0+\sum_{j=1}^N x_jA_j,\qquad A_j\in\mathop{\mathrm{Mat}}_n(\mathbb C), \tag{43}\] for which the same polar system has \(\mu\) distinct simple zeros. This choice constrains only the coefficients of \(Q\). The matrix entries may be arbitrary complex numbers, however large; no bound on an approximation order or on coefficients of a parametrized family is used.

Choose a common chart for the right kernels. At each selected zero, the first row of the polar Jacobian is nonzero, so \(d(Q\circ\Psi)\ne0\). If \(A(\Psi(\xi))\) had rank at most \(n-2\), its adjugate would vanish and the determinant derivative in every direction would be zero. Since its determinant is zero, its rank is exactly \(n-1\). It has unique right and left projective kernel lines.

Choose a linear functional nonzero on every one of the finitely many right kernel lines. Such functionals are the complement of a finite union of proper hyperplanes. A fixed change of basis in the matrix domain makes this functional the first coordinate. Equivalently, replace \(A\) by \(AS\) for a suitable invertible constant matrix \(S\). This multiplies \(Q\) and all its polars by the same nonzero constant and preserves simplicity. Continue to write \(A,Q\) for the resulting matrix and polynomial. Every selected right kernel now has a representative with first coordinate one. Put \[B_i=\partial_{w_i}A=\sum_{j=1}^N(w_i)_j A_j.\] These are constant matrices, and \(\partial_{w_i}Q=\mathop{\mathrm{tr}}(\mathop{\mathrm{adj}}(A)B_i)\).

Lift to a square system of bilinear equations. Let \(A^{\mathrm{hom}}(\xi_0,\xi)\) be the entrywise homogenization of the affine matrix \(A\circ\Psi\), so \(A^{\mathrm{hom}}(1,\xi)=A(\Psi(\xi))\). On \[\mathbb P^{h+1}_{[\xi_0:\xi]}\times\mathbb P^{n-1}_p\times\mathbb P^{n-1}_b\] use the equations \[ A^{\mathrm{hom}} p=0,\qquad (b^{\mathsf T}A^{\mathrm{hom}})_j=0\quad(2\le j\le n),\qquad b^{\mathsf T}B_i p=0\quad(1\le i\le h). \tag{44}\] There are \(n+(n-1)+h=h+2n-1\) equations, equal to the product dimension. In the chart \(p_1\ne0\), the omitted first left-kernel equation follows from the others, since \[0=b^{\mathsf T}(A^{\mathrm{hom}} p) =p_1(b^{\mathsf T}A^{\mathrm{hom}})_1+ \sum_{j=2}^n p_j(b^{\mathsf T}A^{\mathrm{hom}})_j.\] At a selected point, rank \(n-1\) implies \(\mathop{\mathrm{adj}}(A)=\rho p b^{\mathsf T}\) for normalized kernel representatives and some \(\rho\ne0\): every column of the adjugate is in the right kernel, every row is in the left kernel, and some cofactor is nonzero. Consequently \(\partial_{w_i}Q=\rho b^{\mathsf T}B_i p\) there. Each selected affine zero therefore has a unique lift satisfying (44).

The lift is not yet known to be simple. We prove this locally before applying Lemma 22; other components of the kernel system will impose no further requirement.

The same row permutation is applied to every \(B_i\). The last \(n-1\) right equations and all retained left equations give \[\gamma+Eu=0,\qquad \beta+v^{\mathsf T}E=0.\] Their auxiliary Jacobian has invertible diagonal blocks \(E,E^{\mathsf T}\), so they eliminate the kernel coordinates holomorphically as \[ u=-E^{-1}\gamma,\qquad v^{\mathsf T}=-\beta E^{-1}. \tag{45}\] The remaining right equation, evaluated on the slice, is \[s(\xi)=S(\Psi(\xi))=0, \qquad S(x)=a(x)-\beta(x)E(x)^{-1}\gamma(x).\] The Schur determinant identity gives \[ Q(x)=\sigma\det(E(x))S(x), \tag{46}\] where \(\sigma\in\{1,-1\}\) is the sign of the row permutation. It follows by subtracting \(\beta E^{-1}\) times the lower block rows from the first row. Here \(S\) is defined on an ambient neighborhood where \(E\) is invertible, so it can be differentiated in each ambient direction \(w_i\).

After substituting (45), let \(g_i(\xi)\) denote the bilinear equation \(b^{\mathsf T}B_i p\). For any constant ambient direction \(w\), differentiation of \(S\) gives \[\begin{align*} \partial_w S &=\partial_w a-(\partial_w\beta)E^{-1}\gamma +\beta E^{-1}(\partial_w E)E^{-1}\gamma -\beta E^{-1}(\partial_w\gamma)\\ &=(1,-\beta E^{-1})(\partial_w A) \binom{1}{-E^{-1}\gamma}. \end{align*}\] Thus \(g_i=(\partial_{w_i}S)\circ\Psi\). In particular this computation does not identify an ambient derivative with a derivative along the slice. If \(f=Q\circ\Psi\) and \(q_i=(\partial_{w_i}Q)\circ\Psi\), differentiating (46) in the ambient space yields \[ \begin{gathered} f=c\,s,\qquad q_i=c\,g_i+c_i s,\\ c=\sigma(\det E)\circ\Psi\ne0,\qquad c_i=\sigma(\partial_{w_i}\det E)\circ\Psi. \end{gathered} \tag{47}\] The original equation vector \((f,q_1,\ldots,q_h)\) and the reduced vector \((s,g_1,\ldots,g_h)\) therefore differ by a triangular holomorphic matrix with diagonal entries \(c\). At a common zero their Jacobians differ by the value of that invertible matrix. The original zero was simple, so the reduced Jacobian is invertible.

Finally, a tangent vector annihilated by the full kernel-system Jacobian first satisfies the differentials of the eliminated equations. Their invertible auxiliary block determines its \((u,v)\) component from its \(\xi\) component. The remaining equations and the invertible reduced Jacobian force that \(\xi\) component to be zero, and hence the auxiliary component is zero as well. The full square Jacobian is invertible. All \(\mu\) selected lifts are thus distinct simple zeros of (44).

Count the selected lifts. The three groups of equations in (44) have multidegrees and multiplicities \[\begin{array}{c|c|c} \text{equations}&\text{multidegree}&\text{number}\\ \hline A^{\mathrm{hom}} p &(1,1,0)&n\\ (b^{\mathsf T}A^{\mathrm{hom}})_j, j\ge2 &(1,0,1)&n-1\\ b^{\mathsf T}B_i p &(0,1,1)&h. \end{array}\] Lemma 22 gives the first inequality in (41), regardless of other zeros or components. The product has nonnegative coefficients. Setting \(U=V=1\) sums the coefficients with the same \(H\)-exponent, and hence bounds the particular coefficient by \[[H^{h+1}]\,2^h(H+1)^{2n-1} =2^h\binom{2n-1}{h+1}.\] Since \(\mu>0\), we must have \(h+1\le2n-1\). For integers \(1\le b\le a\), \[\binom ab\le\frac{a^b}{b!}\le\left(\frac{ea}{b}\right)^b;\] the last bound follows from \(\log(b!)\ge\int_1^b\log t\,dt\ge b\log b-b\). Taking \((h+1)\)st roots gives \[\mu^{1/(h+1)} \le2^{h/(h+1)}\frac{e(2n-1)}{h+1} \le\frac{4en}{h+1},\] which is (42). No comparison between \(n\) and the number \(N\) of ambient variables, or between \(n\) and \(h+2\), was assumed. ◻

Proof of Theorem 4. Let \(n\) be any size for which \(G\) is a coefficientwise limit of affine \(n\times n\) determinants. Choose the directions and \(L_0\) from Lemma 21. The affine hyperplane \(L_0=1\) identifies with the projective chart \(L_0\ne0\) and contains representatives of all \(\mu=r(r-1)^{d-2}\) selected points. Parametrize it by an affine isomorphism \[\Psi:\mathbb C^{d-1}\longrightarrow\{x\in\mathbb C^d:L_0(x)=1\}.\] Restricting the homogeneous equations to this hyperplane is precisely their dehomogenization in that projective chart. Thus the equations \[G\circ\Psi=0,\qquad (\partial_{w_i}G)\circ\Psi=0\quad(1\le i\le d-2)\] have \(\mu\) distinct simple affine solutions. The directions remain constant vectors in \(\mathbb C^d\); they need not be tangent to \(L_0=1\). Apply Proposition 23 with \(N=d\), \(h=d-2\), and \(P=G\). It gives \[n\ge\frac{d-1}{4e} \bigl(r(r-1)^{d-2}\bigr)^{1/(d-1)} \ge\frac{(r-1)(d-1)}{4e},\] where the last inequality uses \(r\ge r-1>0\). This holds for every admissible size \(n\), so taking the minimum proves the theorem. ◻

Proof. An exact determinant representation is also a constant sequence of approximating determinants. Hence \(\mathop{\mathrm{\overline{dc}}}(f)\le\mathop{\mathrm{dc}}(f)\) for every \(f\), and Theorem 1 applies. ◻

Proof. Let a program with \(N\) vertices have output \(f\). Sum parallel-edge labels and topologically order the vertices. With adjacency convention \(A_{v,u}=\text{label}(u\to v)\), the matrix \(A\) is strictly triangular, \(\det(I-A)=1\), and \((I-A)^{-1}=I+A+\cdots+A^{N-1}\). The path-sum formula and the block determinant identity give \[f=e_t^{\mathsf T}(I-A)^{-1}e_s =\det\begin{pmatrix}I-A&e_s\\-e_t^{\mathsf T}&0\end{pmatrix}.\] This is an affine-linear determinant of size \(N+1\). Padding with isolated vertices covers all programs with at most \(N\) vertices. Taking closures therefore gives \(\mathop{\mathrm{\overline{dc}}}(f)\le N+1\) for every output in the fixed-budget border model, even when the graph varies. After summing parallel edges, there are finitely many graph types for a fixed vertex budget, and each output depends polynomially on the affine-label coefficients. Hence the output set is constructible, so its two closures agree by the argument in Section 2.

For a nonzero output, trim away all vertices and edges outside source-to-sink paths. The remaining underlying graph is connected, so \(N\le E+1\), where \(E\) is its edge count. Thus \(\mathop{\mathrm{\overline{dc}}}(f)\le E+2\). For a fixed-edge-budget approximation to the nonzero permanent, discard any zero outputs and trim each remaining program separately; the same uniform determinantal bound passes to the limit. The finite graph-type argument also applies to this trimmed edge-budget model. Theorem 1 now yields both cubic lower bounds. ◻

Proof. Fix a size \(n\) for which \(g\) belongs to the coefficientwise closure \(\mathcal C_n\) of affine determinants of size \(n\). Then \(\deg g\le n\), and \(\mathcal C_n\) is closed in the space of polynomials of degree at most \(n\). For every fixed \(t\in\mathbb C\setminus\{0\}\), Lemma 2 gives \[t^{-r}g(a+tx)\in\mathcal C_n.\] Writing \(g(a+x)=\sum_{j=r}^{\deg g}G_j(x)\) as a sum of homogeneous terms, the polynomial on the left is \(\sum_{j=r}^{\deg g}t^{j-r}G_j(x)\). It converges coefficientwise to \(G_r\) as \(t\to0\). Closedness therefore gives \(G_r\in\mathcal C_n\). Theorem 4 now bounds this same size \(n\) below by \((r-1)(d-1)/(4e)\). Taking the least admissible \(n\) proves the border assertion, and \(\mathop{\mathrm{dc}}(g)\ge\mathop{\mathrm{\overline{dc}}}(g)\) proves the exact assertion. ◻