Proposition 4.7 — binomial multiplicity bound

Let $K$ be any nontrivially normed field for which there is an isometric field isomorphism either $K\cong\mathbb C$, or $K\cong\mathbb C_p$ for some prime natural number $p$, where $\mathbb C_p$ is the field denoted by $\mathrm{PadicComplex}(p)$. For every embedded product $G$, every analytic datum $A$, every connected algebraic subgroup $H$, every $g\in G$, every ideal $I\subseteq R$ closed under all block-homogeneous component projections, and every $T\in\mathbb N$, put $V=g+H$. Assume that $V=\{x\in G:P(x)=0\text{ for every }P\in J_V\}$ and that every minimal prime $\mathfrak q$ of $J_V$ whose equations vanish at some point of $G$ is a minimal prime of $J_G+I$. Assume the same two requirements with $I$ replaced by $\mathcal D_A^T(I)$. Then those requirements with the original $I$ hold, and for every prime $\mathfrak q$ minimal over $J_V$ whose equations vanish at some point of $G$, the length of $R_{\mathfrak q}/(J_G+I)R_{\mathfrak q}$ as an $R_{\mathfrak q}$-module is at least $\binom{T+a_A(H)}{a_A(H)}$. Length takes values in $\mathbb N\cup\{\infty\}$; it is zero for the zero module and $\infty$ for a module of infinite length. The codimension in the binomial is computed from the subgroup $H$ through zero, whereas the component conditions concern $V=g+H$. The differential ideal has translation parameter $0$, regardless of this $g$. An embedded product $G$ consists of a positive number $s$ of factors $E_i$; each factor is a locally closed subset of $\mathbb P^{N_i}(K)$, with $N_i\in\mathbb N$, carrying an abelian additive group structure. Its addition and negation are required to have local projective polynomial representations: at every source point and for each target block, some open neighborhood in the source projective Zariski topology and some tuple of homogeneous polynomials of a common block multidegree represent that block of the map at every source point in the neighborhood, and the evaluated tuple is nonzero there. The points of $G$ are the Cartesian product of the factor carriers, and $R=K[X_{ij}:1\leq i\leq s,\ 0\leq j\leq N_i]$ is its coordinate polynomial ring. In each projective block a nonzero representative is chosen; $P(x)$ denotes evaluation at these chosen representatives. A polynomial is multihomogeneous of degree $D$ when every monomial in its support has total exponent $D_i$ in block $i$; the zero polynomial has every such degree. The ambient Zariski topology is generated by the sets where a multihomogeneous polynomial is nonzero, and $G$ has the induced topology. For any $V\subseteq G$, $J_V\subseteq R$ is the ideal generated by all multihomogeneous polynomials vanishing at every point of $V$; write $J_G$ for the whole group and $Z(P)=\{x\in G:P(x)=0\}$. An algebraic subgroup means an additive subgroup closed in this topology; connectedness refers to this topology. Translation means $g+V=\{g+v:v\in V\}$. The data $A$ consist of a natural number $d>0$, a real radius $\rho>0$, the open ball $B=\{z\in K^d:\|z\|<\rho\}$ with its asserted openness and membership $0\in B$, and a map $\phi:B\to G$ such that $\phi(0)=0$ and $\phi(x+y)=\phi(x)+\phi(y)$ whenever $x,y,x+y\in B$. No injectivity is required. There are coordinate lifts $L_g(z)_{ij}$, defined for every $g\in G$ and $z\in K^d$, each analytic at $0$; for $g=0$ each coordinate admits a formal multilinear power series on the entire radius-$\rho$ ball and each block of $L_0(z)$ is nonzero and represents $\phi(z)$ for every $z\in B$. For every $g$, on some neighborhood of $0$ contained in $B$, every block of $L_g(z)$ is nonzero and represents $g+\phi(z)$. The set $A_*\subseteq G$ is the additive subgroup generated by $\phi(B)$, with no topological closure operation. Put $F_{P,g}(z)=P(L_g(z))$ and $K_A(V)=\bigcap_{P\in J_V}\ker(D F_{P,0}(0))\subseteq K^d$. The natural number $a_A(V)$ is $d-\dim_K K_A(V)$, using natural-number subtraction, and $\dim A$ is $a_A(\{0\})$. The order $o_A(P,g)$ is the least $k\in\mathbb N$ for which the $k$-fold Fréchet derivative of $F_{P,g}$ at $0$ is nonzero, or $\infty$ if all these derivatives vanish; the derivative of order zero is the value at $0$. Derivatives use the total Fréchet derivative, which is zero where differentiability fails. All order comparisons take place in $\mathbb N\cup\{\infty\}$. For a chart $b=(b_i)$, $0\leq b_i\leq N_i$, let $U_b=\{x\in G:x_{i,b_i}\ne0\text{ for every }i\}$, where $x_{ij}$ are chosen projective representatives, and let $Q_b(x)=Q((x_{ij}/x_{i,b_i})_{ij})$. The differential ideal $\mathcal D_A^T(I)$ used here has translation parameter zero. Its generating local sections are obtained from every multihomogeneous $P\in I$, every chart $b$, every $0\leq k\leq T$, and every ordered list of $k$ coordinate directions in $K^d$: the section has domain $U_b$ and value at $x$ equal to the $k$-fold Fréchet derivative at $z=0$ of $P((L_x(z)_{ij}/L_x(z)_{i,b_i})_{ij})$, applied to those standard coordinate vectors. Repetitions of directions are allowed and $k=0$ means the function value, with no factorial factor. To form $\mathcal D_A^T(I)$, take every multihomogeneous $Q$ with this property at every $x\in G$: there are a chart $b'$ and a Zariski open neighborhood $U$ of $x$ contained in $U_{b'}$ such that $Q_{b'}$ on $U$ is a finite sum $\sum_\nu r_\nu f_\nu$ of the foregoing local sections, each defined on all of $U$, with each coefficient $r_\nu$ the ratio of two homogeneous polynomials of the same block multidegree and its denominator nonzero everywhere on $U$. Then take the polynomial ideal spanned by all these $Q$. The finite sum may be empty. All rational functions and normalized pullbacks are defined globally using field division, including value zero for a quotient with denominator zero, but the local membership representations explicitly exclude such denominators on their neighborhoods. The component condition uses all minimal primes meeting actual group points, with no independent relevance or projective-open test. Primes not meeting $G$ impose no component or length requirement, and the quantifiers over eligible primes are vacuous if there are none. The natural parameter $T=0$ is allowed, in which case only order-zero sections occur and the asserted lower bound is $1$; the lower bound is also $1$ whenever $a_A(H)=0$. The parameter dimension is positive, but the defined analytic dimension and codimension may be zero. No nonzero, properness, or bounded-generator-degree hypothesis is imposed on $I$.