Theorem 2.1 — general multiplicity estimate

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 $\mathrm{PadicComplex}(p)$. There exists a function $c$ assigning a natural number to every embedded commutative group over $K$, with $c(E)\geq1$ for every such $E$, such that the following holds simultaneously for every embedded product $G=\prod_iE_i$, every analytic datum $A$ on $G$, every finite $\Sigma\subseteq G$ containing $0$, every $T\in\mathbb N$, every $D\in\mathbb N^s$, and every nonzero $P\in R$ multihomogeneous of degree exactly $D$. If $o_A(P,g)\geq nT+1$ for every $g\in\Sigma^{[n]}$, then there is a connected algebraic subgroup $H$ of $G$ having an incomplete definition of degrees at most $(c(E_i)D_i)_i$, such that $H\subseteq g_0+Z(P)$ for some $g_0\in G$, and $\binom{T+a_A(H)}{a_A(H)}C_\Sigma(H)F_H(D)\leq F_G((c(E_i)D_i)_i)$. The translation clause means that each $h\in H$ can be written $h=g_0+z$ with $P(z)=0$. The function $c$ is chosen before $G,A,\Sigma,T,D,P$, and a factor receives the same assigned value wherever that identical embedded group occurs. 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 any ideal $J\subseteq R$ and block degree $e\in\mathbb N^s$, let $h_J(e)$ be the $K$-dimension of the image of the vector space of multihomogeneous polynomials of degree $e$ in $R/J$. Let $Q_J\in\mathbb Q[t_1,\ldots,t_s]$ be a polynomial agreeing with $h_J(e)$ at every natural vector $e$ above some coordinatewise natural threshold, if such a polynomial exists; if none exists, set $Q_J=0$. The polynomial, when it exists, is unique. Set $\delta(J)=\deg Q_J$, with total degree of the zero polynomial defined as $0$, and $F_J(D)=\delta(J)!\,[Q_J]_{\delta(J)}(D)$, where brackets select the total homogeneous part of that degree. This is a rational value, viewed as a real number in the group inequalities. Write $\delta(V)=\delta(J_V)$ and $F_V(D)=F_{J_V}(D)$. The number $n_i$ assigned to a factor $E_i$ is the total degree of the corresponding polynomial $Q$ for its vanishing ideal in its one-block homogeneous coordinate ring, and $n=\sum_i n_i$. None of these definitions supplies a separate existence hypothesis for $Q_J$; the zero fallback makes both $\delta(J)$ and $F_J$ equal to zero in that case. Evaluation permits zero coordinates of $D$, with $0^0=1$. Saying that $V$ has an incomplete definition of degrees at most $B=(B_i)$ means that there is an ideal $I\subseteq R$ closed under every block-homogeneous component projection, and a finite set $\mathcal E\subseteq R$ whose members are each homogeneous of some degree $e\leq B$ coordinatewise, such that $J_G+I=J_G+(\mathcal E)$, $V=\{x\in G:P(x)=0\text{ for every }P\in J_V\}$, and every minimal prime $\mathfrak q$ of $J_V$ whose equations vanish at some point of $G$ is a minimal prime of $J_G+I$. The finite equation set may be empty and may contain zero; components not meeting $G$ impose no condition, and additional minimal primes of $J_G+I$ are not excluded. For a finite set $\Sigma\subseteq G$, $\Sigma^{[n]}$ is the set of sums of exactly $n$ members of $\Sigma$, allowing repetitions, so $\Sigma^{[0]}=\{0\}$. The number $C_\Sigma(H)$ counts distinct subsets $g+H$ for $g\in\Sigma$, not elements of a multiset. The quantifiers include $T=0$, zero entries of $D$, factors with $n_i=0$, and $n=0$; they exclude an empty factor list and, because $0\in\Sigma$, an empty sample. If $n=0$, the vanishing hypothesis is imposed only at $0$ and has threshold $1$. If $T=0$ or $a_A(H)=0$, the binomial factor is $1$. A nonzero polynomial in $R$ is allowed to vanish on every point of $G$ unless a separate nonvanishing-on-$G$ clause is stated.