Lemma 5.1 — stabilizer and geometric counting

For every field $K$ with a nontrivial norm, assume that either there is an isometric ring isomorphism $K\simeq\mathbb C$, or there are a natural prime $p$ and an isometric ring isomorphism from $K$ to $\mathbb C_p$, the completion of the algebraic closure of $\mathbb Q_p$ with its specified $p$-adic norm. The common data are a field $K$ with a nontrivial norm and a product $G=\prod_{i=1}^{m}G_i$ with $m>0$. Each $G_i$ is a subset of $\mathbf P(K^{N_i+1})$, where $N_i\in\mathbb N$ may be zero, equipped with an additive commutative group law; this subset is locally closed in the projective Zariski topology, and addition and negation have local polynomial presentations. More explicitly, at every input and for each output projective factor there is a Zariski open neighborhood in the input projective space and a tuple of input multihomogeneous polynomials of one common multidegree whose evaluated vector is nonzero and represents that output at every input in the neighborhood. Put $R=K[X_{i,j}:1\le i\le m,\ 0\le j\le N_i]$, and let $r(x)_{i,j}$ be the coordinate of the fixed nonzero representative chosen for the projective point $x_i$; evaluation at a group point means $P(r(x))$. A polynomial is multihomogeneous of multidegree $D\in\mathbb N^m$ when every monomial in its support has sum of exponents $D_i$ in block $i$; the zero polynomial satisfies this condition for every $D$. The Zariski topology on the ambient product is generated by the sets where one such polynomial evaluates nonzero, and $G$ has the topology induced by its inclusion. For $V\subseteq G$, write $\mathcal I(V)$ for the ideal of $R$ generated by all multihomogeneous polynomials, of any multidegree, whose evaluations $P(r(x))$ vanish for every $x\in V$, and write $I_G=\mathcal I(G)$. In particular, this is an ideal span of homogeneous vanishing polynomials. The analytic datum $A$ consists of an integer $d>0$, a real radius $\rho>0$, the open ball $B=\{z\in K^d:\|z\|<\rho\}$ about zero (with its specified openness and membership $0\in B$), and a map $\phi:B\to G$ with $\phi(0)=0$ and $\phi(x+y)=\phi(x)+\phi(y)$ whenever $x,y,x+y\in B$. It also contains total coordinate functions $L_g(z)_{i,j}\in K$ for every $g\in G$ and $z\in K^d$. Every such coordinate function is analytic at $z=0$. For each coordinate, $L_0$ has a formal multilinear power series representing it on the ball of radius $\rho$; for every $z\in B$, each vector $L_0(z)_i$ is nonzero and represents $\phi(z)_i$. For every $g\in G$, on some neighborhood of $0$ all parameters belong to $B$, each vector $L_g(z)_i$ is nonzero, and its projective point is $(g+\phi(z))_i$. These latter neighborhoods may depend on $g$. Injectivity of $\phi$ and positivity of its derivative rank are not hypotheses. The parameter space is the full normed vector space $K^d$, including points outside $B$. Here $D^k f(0)$ denotes the $k$th iterated Fréchet derivative over $K$ as a continuous $k$-linear map. Its value for $k=0$ is $f(0)$, with an empty tuple of arguments. At each positive recursive step the Fréchet derivative is defined to be zero if the differentiated function is not differentiable there; these expressions are total. Write $e_a$ for the vector with coordinate $a$ equal to $1$ and every other coordinate equal to $0$. For $g\in G$, a translation chart consists of a Zariski open set $U\subseteq G$, a degree vector $\delta\in\mathbb N^m$, and polynomials $F_{i,j}\in (K^{K^d})[X]$. Every coefficient function of each $F_{i,j}$ that is nonzero as a function is analytic at $0$. Every supported monomial of $F_{i,j}$ has degree $\delta_i$ in block $i$ and degree zero in each other block. Set $E_{i,j}(x,z)=F_{i,j}(r(x),z)$, evaluating coefficient functions at $z$ and polynomial variables at $r(x)$. For all $x\in G$, all $i$, and all $j,k$ in block $i$, the equality $E_{i,j}(x,z)L_{g+x}(z)_{i,k}=E_{i,k}(x,z)L_{g+x}(z)_{i,j}$ holds on some neighborhood of $0$; the neighborhood may depend on $x,i,j,k$. For every $x\in U$, on some neighborhood of $0$ one has $z\in B$ and each vector $(E_{i,j}(x,z))_j$ is nonzero and represents $(x+g+\phi(z))_i$. An atlas for $g$ is a family of such charts indexed by an arbitrary type, together with the requirement that every $x\in G$ belongs to at least one chart domain. Individual chart domains may be empty, and chart degrees may be zero; the coefficient and monomial conditions on a zero coordinate polynomial are vacuous. The atlas index type cannot be empty, because $G$ contains zero. A bound $c\in\mathbb N^m$ means $\delta_{a,i}\le c_i$ for every chart $a$ and factor $i$. For a chart $a$ and $P\in R$, substitute its polynomials $F_{a,i,j}$ into $P$, embedding each scalar coefficient of $P$ as a constant function on $K^d$, and write the resulting finite polynomial as $\sum_{\alpha}b_\alpha(z)X^\alpha$. For $k\in\mathbb N$ and an ordered tuple $v:\{0,\ldots,k-1\}\to\{0,\ldots,d-1\}$, define $\mathcal D_{a,k,v}P=\sum_{\alpha}D^k b_\alpha(0)(e_{v(0)},\ldots,e_{v(k-1)})X^\alpha$. This sum runs over the finite support of the coefficient function polynomial, differentiates the coefficients only, and has no factorial divisor. Directions may repeat. At $k=0$ the result is the substituted polynomial with coefficient functions evaluated at $0$. For any ideal $J$ in one of these block polynomial rings, let $h_J(a)$ be the dimension over $K$ of the image, in $R/J$, of the vector space of polynomials homogeneous of exact block degree $a\in\mathbb N^m$. Define $\mathscr H_J\in\mathbb Q[t_1,\ldots,t_m]$ by choosing a polynomial for which there exists $a_0\in\mathbb N^m$ such that $\mathscr H_J(a)=h_J(a)$ for every $a$ with $a_i\ge(a_0)_i$ for all $i$, if any such polynomial exists; if none exists, define $\mathscr H_J=0$. Its total degree is the largest sum of monomial exponents in its support, with total degree of zero defined to be zero. Put $\dim_G(V)=\deg_{\mathrm{tot}}\mathscr H_{\mathcal I(V)}$. The number $n=\dim(G)$ used below is, specifically, the sum over $i$ of the total degrees of the analogous one-block Hilbert polynomials for the homogeneous vanishing ideals of the individual subsets $G_i\subseteq\mathbf P(K^{N_i+1})$; it is a natural number and may be zero. For $P\in R$ and $g\in G$, the order $\operatorname{ord}_{A,g}(P)$ is the infimum in $\mathbb N\cup\{\infty\}$ of the natural numbers $k$ for which the entire multilinear map $D^k(z\mapsto P(L_g(z)))(0)$ is nonzero. Thus it is the least such $k$ if one exists, and is $\infty$ if all these derivatives are zero; order zero tests $P(L_g(0))$. For every such $G$ and $A$, assume a supplied construction $C$ with all of the following data and conditions. The input consists of a finite subset $\Sigma\subseteq G$ with $0\in\Sigma$, a natural number $T$, a multidegree vector $D\in\mathbb N^m$, a polynomial $P\in R$ which is nonzero as an element of $R$ and is multihomogeneous of degree $D$, a vector $c\in\mathbb N^m$ with $1\le c_i$ for every $i$, and, for every $g\in G$, a translation atlas bounded coordinatewise by $c$ as described above. If $\Sigma(q)$ denotes the finite set of sums of exactly $q$ elements of $\Sigma$, with repetitions permitted, then $\Sigma(0)=\{0\}$ and the input requires $nT+1\le\operatorname{ord}_{A,g}(P)$ for every $g\in\Sigma(n)$, with the natural number on the left included in $\mathbb N\cup\{\infty\}$. Thus the threshold is $1$ when either $n=0$ or $T=0$, and infinite order satisfies the inequality. Individual $D_i$ may be zero, and nonzeroness of $P$ does not require that $P$ be nonzero as a function on $G$. If every $D_i=0$, multihomogeneity and $P\ne0$ make $P$ a nonzero constant, whose order is zero; since $0\in\Sigma(n)$, that choice cannot satisfy the contact requirement. The sequence of ideals is $I_0=I_G$, $I_1=I_G+(P)$, and, for every $q\in\mathbb N$, $I_{q+2}=I_G+\langle\mathcal D_{a,k,v}P:\ g\in\Sigma(q+1),\ a\text{ is a chart of the atlas for }g,\ 0\le k\le(q+1)T,\ v\in\{0,\ldots,d-1\}^{k}\rangle$. These generators use every chart, including any chart with empty domain; there is no condition that a group point belong to that chart domain when the generator is included. The displayed ideals use ordinary ideal generation in $R$. At $T=0$ only order-zero operators occur in the terms with index at least $2$. A construction additionally selects an index $r\in\mathbb N$ with $r\le n$ and requires $\dim_G(Z_G(I_r))=\dim_G(Z_G(I_{r+1}))$, where $Z_G(J)=\{x\in G:\forall Q\in J,\ Q(r(x))=0\}$. It selects a proper prime ideal $\mathfrak p$ of $R$ that is minimal among prime ideals containing $I_r$ and is also minimal among prime ideals containing $I_{r+1}$. There must exist $x\in G$ at which every polynomial in $\mathfrak p$ evaluates zero. Put $V=Z_G(\mathfrak p)$; the construction requires $\dim_G(V)=\dim_G(Z_G(I_r))$ and requires the additive subgroup $S=\{h\in G:h+V=V\}$ to be closed in the specified Zariski topology on $G$. Here $h+V=\{h+x:x\in V\}$, and $r=0$ is allowed, including when $n=0$. The prime is not separately required to be multihomogeneous, and its meeting condition forces $V$ to be nonempty. This structure includes every field and condition of the input just described. To specify the ideals in the conclusions, a coordinate chart $b$ chooses one index $b_i\in\{0,\ldots,N_i\}$ in every block. Set $U_b=\{x\in G:\forall i,\ r(x)_{i,b_i}\ne0\}$ and $q_b(Q,x)=Q((r(x)_{i,j}/r(x)_{i,b_i})_{i,j})$. A local section consists of a subset of $G$ as domain and a total function $G\to K$ as value. For any collection $\mathscr S$ of such sections, define $\mathcal L(\mathscr S)$ as the ideal generated by the multihomogeneous polynomials $Q\in R$ of any degree satisfying the following condition at every $x\in G$: there exist a coordinate chart $b$, a Zariski open set $U$ with $x\in U\subseteq U_b$, an integer $\ell\ge0$, sections $f_1,\ldots,f_\ell\in\mathscr S$, and polynomials $a_j,b_j\in R$ such that, for each $j$, the numerator $a_j$ and denominator $b_j$ are multihomogeneous of the same multidegree (which may depend on $j$), every $y\in U$ belongs to the domain of $f_j$, $b_j(r(y))\ne0$ for every $y\in U$, and $q_b(Q,y)=\sum_{j=1}^{\ell}(a_j(r(y))/b_j(r(y)))f_j(y)$ for every $y\in U$. The integers, charts, open sets, sections, degrees, and fractions may all depend on $x$ and $Q$. The case $\ell=0$ is permitted and makes the sum zero. The ideal is the span of the homogeneous polynomials with this property, so membership of an arbitrary polynomial in $\mathcal L(\mathscr S)$ is membership in that ideal span. Fraction values and chart values are total expressions, with division by zero defined as zero, but the neighborhood condition explicitly excludes zeros of the selected denominators and of the chart coordinates on $U$. For $g\in G$, $t\in\mathbb N$, and an ideal $I\subseteq R$, take one section for every homogeneous $Q\in I$ (including $Q=0$), coordinate chart $b$, integer $0\le k\le t$, and ordered direction list $v\in\{0,\ldots,d-1\}^k$. Its domain is $\{x\in G:g+x\in U_b\}$, and its total value at $x$ is $D^k\bigl(z\mapsto Q((L_{g+x}(z)_{i,j}/L_{g+x}(z)_{i,b_i})_{i,j})\bigr)(0)(e_{v_0},\ldots,e_{v_{k-1}})$. Denote the ideal $\mathcal L$ generated in the preceding local sense from these sections by $\mathfrak D_{g,t}(I)$. Repeated coordinate directions and order zero are included, there is no factorial divisor, and if $t=0$ only the empty direction list and zeroth derivative are used. For the translated ideal $\mathfrak T_g(I)$, instead take one section for every homogeneous $Q\in I$ and coordinate chart $b$, with domain $\{x:g+x\in U_b\}$ and total value $q_b(Q,g+x)$, and apply the same $\mathcal L$ operation. Thus both constructions use local equations throughout $G$, and the translation in the section formulas is by addition of $g$. The theorem asserts the conjunction of the following three conclusions. First, for every $g\in\Sigma$ one has the ideal inclusion $\mathfrak D_{g,T}(I_r)\subseteq\mathfrak p$ in the ordinary polynomial ring $R$. Second, let $H$ be the connected component of $0$ in the subspace $S$ for the specified Zariski topology, viewed as a subset of $G$; equivalently, since $0\in S$, it is the union of the connected subsets of $S$ containing $0$. The general connected-component-in-a-set operation returns the empty set when the point is outside the set, but that alternative does not apply here because $S$ is a subgroup. Put $W_H=\bigcap_{Q\in\mathcal I(H)}\ker(D(z\mapsto Q(L_0(z)))(0))\subseteq K^d$ and $s=d-\dim_K W_H$, where the derivative kernels are linear subspaces and subtraction is natural-number subtraction, truncated at zero. In this definition every polynomial in $\mathcal I(H)$ is used, the pullback uses the unnormalized lift $L_0$, and $s=0$ is allowed. Let $c_H=\#\{g+H:g\in\Sigma\}$, counting distinct translated subsets, so different sampling points producing the same subset are counted once. For any $Y\subseteq G$ and $E\in\mathbb N^m$, define $F_Y(E)$ by taking $\delta_Y=\deg_{\mathrm{tot}}\mathscr H_{\mathcal I(Y)}$, keeping the part of this polynomial consisting of monomials of total degree exactly $\delta_Y$, multiplying that polynomial by the single factorial $\delta_Y!$, evaluating at the natural vector $E$ in $\mathbb Q$, and then including the result in $\mathbb R$. If the selected Hilbert polynomial is zero, this value is zero; coordinates of $E$ may be zero. With $D'_i=c_iD_i$, the exact real inequality is $\binom{T+s}{s}\,c_H\,F_H(D)\le F_G(D')$, where the natural binomial coefficient and the natural cardinality $c_H$ are cast to real numbers. When $s=0$ or $T=0$ the binomial coefficient is $1$; no strict inequality or positive degree-form hypothesis is present. Third, let $J=\sum_{v\in V}\mathfrak T_v(I_r)$, meaning the least ideal containing every translated ideal indexed by every point $v$ of $V$, with the arbitrary, possibly infinite ideal sum taken in $R$. The theorem asserts both the equality $S=\{x\in G:\forall Q\in\mathcal I(S),\ Q(r(x))=0\}$ and the following condition: for every prime ideal $\mathfrak q$ minimal among primes containing $\mathcal I(S)$, if there exists $x\in G$ with $Q(r(x))=0$ for every $Q\in\mathfrak q$, then $\mathfrak q$ is also minimal among prime ideals containing $I_G+J$. Minimality is with respect to ideal inclusion, not merely primality or containment. This last conclusion concerns the full stabilizer $S$, whereas the numerical inequality uses $H$. Its prime condition is an implication only for minimal primes meeting $G$; if there were no such primes, that universal implication would be vacuous while the displayed set equality would still be required. The sum is indexed by all points of the selected component $V$, which is nonempty by hypothesis.