1987 addendum — converse construction (positive dimension)

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 $G$ satisfying $n=\sum_i n_i>0$, every analytic datum $A$, every finite $\Sigma\subseteq G$ containing $0$, every $T\in\mathbb N$, every $D\in\mathbb N^s$ satisfying $F_G(\mathbf1)\leq D_i$ in $\mathbb R$ for every $i$, and every algebraic subgroup $H$, assume $\binom{T+a_A(H)}{a_A(H)}C_\Sigma(H)F_H(D)\leq\bigl(4^n n!\bigr)^{-1}F_G(D)$. Then there exists a polynomial $P\in R$ such that $P\ne0$, $P$ is multihomogeneous of degree exactly $D$, $o_A(P,g+h)\geq T+1$ for every $g\in\Sigma$ and $h\in H$, and $P(x)\ne0$ for at least one $x\in G$. Here $\mathbf1$ is the vector having entry $1$ in every block, and the lower bounds on the entries of $D$ are real comparisons with that single degree-form value. The subgroup $H$ need not be connected. The nonvanishing conclusion is an additional condition on restriction to $G$ as well as nonzeroness in the polynomial ring. 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 number $C_\Sigma(H)$ counts distinct subsets $g+H$ for $g\in\Sigma$, not elements of a multiset. The condition $n=\sum_i n_i>0$ excludes $n=0$ and requires at least one factor of positive defined dimension, while individual factors with $n_i=0$ are allowed. The natural values $T=0$ and $D_i=0$ are included whenever the remaining hypotheses hold; the inequalities on $D$ do not separately assume $D_i>0$. Since $n\geq1$, the factor $4^n n!$ is at least $4$, so its reciprocal is positive and at most $1/4$, with no zero-denominator case. The order threshold includes the order-zero derivative, and for $T=0$ it is exactly $1$. Because $0\in\Sigma$, the vanishing conclusion holds at every point of $H$; hence a polynomial satisfying both conclusion clauses cannot occur when $H=G$. The sample cannot be empty, $0^0=1$ in degree evaluation, and the Hilbert-polynomial zero fallback remains part of every degree-form value in the hypotheses.