Section 2 — recovery of Masser–Wüstholz Theorem I

Let $K$ be any nontrivially normed field for which there is either an isometric ring isomorphism $K\cong\mathbb C$, or a prime natural number $p$ and an isometric ring isomorphism $K\cong\operatorname{PadicComplex}(p)$. An embedded group product $G$ consists of a positive finite number $s$ of commutative groups $E_i$, each carried by a locally closed subset of $\mathbb P^{N_i}(K)$, with addition and negation locally represented by nonsimultaneously-zero homogeneous polynomial tuples of a common multidegree; its points are the tuples of factor points, with componentwise addition, embedded in $M=\prod_i\mathbb P^{N_i}(K)$. Write $R=K[X_{ij}]$ for the block coordinate ring, evaluate polynomials at the chosen homogeneous representatives of embedded points, and use the induced Zariski topology, where the ambient topology is generated by nonvanishing sets of block-homogeneous polynomials. A polynomial is homogeneous of multidegree $D$ when every supported monomial has degree $D_i$ in block $i$; zero is allowed at every multidegree. An algebraic subgroup means an additive subgroup closed in this topology, and $g+V=\{g+x:x\in V\}$. For $V\subseteq G$, let $\mathcal I(V)$ be the ideal spanned by all homogeneous polynomials vanishing on its embedded points, and let $H_V$ be the rational polynomial whose values, for all coordinatewise sufficiently large natural multidegrees, are the dimensions of the images of those homogeneous pieces in $R/\mathcal I(V)$, chosen if it exists and set to zero if none exists. Write $\delta(V)=\deg_{\mathrm{tot}}H_V$ (zero for the zero polynomial), $\deg_V(D)=\delta(V)!\,(H_V)_{\delta(V)}(D)\in\mathbb R$, and $\mu_V(\alpha)=[t^\alpha]H_V\prod_i\alpha_i!\in\mathbb R$ if $\sum_i\alpha_i=\delta(V)$, with $\mu_V(\alpha)=0$ otherwise. Here $D,\alpha$ are natural multidegrees and $(H_V)_{\delta(V)}$ is the top total-homogeneous component. For each factor, $d_i$ is the same Hilbert-polynomial total degree computed for its carrier in its single projective space, and $n=\sum_i d_i$ is the defined dimension of $G$. Take any embedded commutative group $E\subseteq\mathbb P^N(K)$ and let $G$ be its one-factor product, so $n=d_E$; assume $n>0$ and that the whole point set of $G$ is connected. Let natural numbers $a,b$ satisfy $a,b\ge1$. Assume that for every finitely generated $\mathbb Z$-submodule $\Gamma\subseteq G$ and every $g\in\Gamma$ there is one Zariski-open polynomial translation chart containing all of $\Gamma$, of degree at most $a$: its homogeneous coordinate polynomials all have a common natural degree at most $a$, and at every point of its domain their tuple is nonzero and represents $g+x$. Also assume that a finite set of homogeneous coordinate polynomials, each of some degree at most $b$, cuts out exactly the ambient projective Zariski closure of $\iota(G)$. Let $m,D\in\mathbb N$ satisfy $m,D\ge1$, let $\gamma_0,\ldots,\gamma_{m-1}$ be arbitrary points of $G$, let $\theta\in\mathbb R$ satisfy $n/m\le\theta$, and let $P\in R$ be homogeneous of some natural degree at most $D$. Define $c=1/(a^n b^{N-n})\in\mathbb R$, where $N-n$ is natural subtraction, and put $\Lambda_\gamma(S)=\{\sum_{i<m}u_i\gamma_i:u_i\in\mathbb N,\ u_i\le S\text{ for every }i\}$. If $P$ vanishes at every embedded point of $\Lambda_\gamma(n(D/c)^\theta)$ and there exists at least one $x\in G$ at which $P$ does not vanish, then there exist natural numbers $k,r$ satisfying $1\le k\le m$, $1\le r\le n$, and the strict real inequality $m<k+r/\theta$; a $\mathbb Z$-submodule $Z\subseteq\mathbb Z^m$ with $k\le\operatorname{finrank}_{\mathbb Z}Z$; and an algebraic subgroup $H$ such that $\delta(H)\le n-r$ and $\sum_{i<m}\sigma_i\gamma_i\in H$ for every $\sigma\in Z$. Moreover there are $k$ elements $\sigma^{(0)},\ldots,\sigma^{(k-1)}\in Z$ whose images in $\mathbb Z^m$ are linearly independent over $\mathbb Z$, with $|\sigma_i^{(j)}|\le(D/c)^{\,r/(m-j)}$ for every $0\le j<k$ and every $0\le i<m$. Finally there is a locally closed subset $S\subseteq G$ containing $H$ with $\delta(S)\le n-r$ and a finite set of homogeneous coordinate polynomials cutting out exactly $S$ within $G$, each polynomial having some natural degree at most the real number $D/c$. The subgroup $H$ is not required to be connected, $Z$ is not required to have rank exactly $k$ or to be saturated, and $S$ is not required to be irreducible. The stated hypotheses force $c>0$ and $\theta>0$, and $j<k\le m$ makes every denominator $m-j$ positive; natural subtraction in $N-n$ is defined even without a separately stated inequality $n\le N$. Nonvanishing at one group point is an explicit hypothesis, stronger than nonzeroness merely as a coordinate polynomial.