Corollary 2.3 — disjoint group factors

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 $G=\prod_iE_i$ satisfying the factor condition specified below, there exists a real number $c>0$ such that for every analytic datum $A$, every $l\in\mathbb N$, every family $\gamma=(\gamma_j)_{j=1}^l$ of points of $G$, every real $S\geq0$, every $T\in\mathbb N$, every $D\in\mathbb N^s$, and every nonzero polynomial $P\in R$ multihomogeneous of degree exactly $D$, the following two assumptions imply that $g_0+A_*\subseteq Z(P)$ for some $g_0\in G$. The first assumption is $o_A(P,g)\geq nT+1$ for every $g$ of the form $\sum_{j=1}^l a_j\gamma_j$ with $a_j\in\mathbb N$ and $a_j\leq nS$ for every $j$. The second assumption is that for every natural vector $r$ with $r_i\leq n_i$ for every $i$, one has $c\prod_i D_i^{r_i}\leq (T+1)^{\alpha_A(r)} S^{\beta_{A,\gamma}(r)}$. This quantifier on $r$ has no additional sum restriction. The number $c$ may depend on $K$ and the whole product $G$ and is chosen before $A,l,\gamma,S,T,D,P$. 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 $0^0=1$. The condition on the factors means that every algebraic subgroup $H\subseteq G$, with no connectedness restriction, is exactly the product of its coordinate images $H_i=\{x_i:x\in H\}$, and each $H_i$ is closed in the topology induced on $E_i$ from its one-factor projective Zariski topology. For any algebraic subgroup, put $b_i(H)=n_i-\delta_i(H_i)$, where $\delta_i(H_i)$ is the total degree of the eventual Hilbert polynomial of the homogeneous vanishing ideal of the image $H_i$ in that one-factor projective space, using zero as the polynomial if none exists; the subtraction is truncated at zero. For a natural vector $r$, consider all algebraic subgroups $H$ such that $A_*\nsubseteq H$ and $r_i\leq b_i(H)$ for every $i$. Connectedness is not part of this condition. Let $\alpha_A(r)$ be the minimum of $a_A(H)$ over this family, and let $\beta_{A,\gamma}(r)$ be the minimum, separately over the same family, of the $\mathbb Z$-rank of the subgroup generated by the images of $\gamma_1,\ldots,\gamma_l$ in $G/H$. Each minimum is defined to be $0$ when the family is empty; the two minima need not be attained by the same $H$. The rank used is the natural-valued dimension of the $\mathbb Z$-span of those quotient images, namely the supremum of cardinalities of $\mathbb Z$-linearly independent subsets converted to a natural number, with value zero if that supremum is infinite; torsion contributes no rank, and in this finitely generated span the rank is finite. The sampled coefficients are nonnegative natural numbers; negative coefficients are not included. The possibilities $l=0$, $S=0$, $T=0$, $D_i=0$, $n_i=0$, $n=0$, repeated $\gamma_j$, and $\gamma_j=0$ are included, and no positive value of $\dim A$ is required. For $l=0$ the sampling set is $\{0\}$ and every sampling quotient rank is zero. For $S=0$ or $n=0$, the sampling set is also $\{0\}$. Every zero base raised to exponent zero is $1$; thus an empty obstruction family makes the right-hand side of the corresponding inequality equal to $1$, including when $S=0$. The conclusion concerns the entire subgroup generated by the local image, and $P$ is not required to be nonzero at a group point.