Corollary 2.2 — one-dimensional analytic subgroup (positive degrees)

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 $G$. An analytic subgroup datum $A$ has a positive natural parameter dimension $q$, a positive real radius $\rho$, the open ball $B(0,\rho)\subset K^q$, and a map $a:B(0,\rho)\to G$ with $a(0)=0$ and $a(u+v)=a(u)+a(v)$ whenever $u,v,u+v$ lie in that ball. It also supplies functions $\ell_g:K^q\to K^{\{(i,j)\}}$ for every $g\in G$: every coordinate is analytic at zero, the coordinates of $\ell_0$ have convergent power series on the whole ball and give nonsimultaneously-zero projective representatives of $a(z)$ there, and, as germs at zero, the nonzero block tuples of $\ell_g(z)$ represent $g+a(z)$. No injectivity of $a$ is required. The carrier $A^\ast$ is the additive subgroup generated by $a(B(0,\rho))$, not a topological closure. For $V\subseteq G$, put $T_A(V)=\bigcap_{P\in\mathcal I(V)}\ker d_0(z\mapsto P(\ell_0(z)))$ and $\kappa_A(V)=q-\dim_K T_A(V)$, using natural subtraction; the defined dimension of $A$ is $\kappa_A(\{0\})$. Write $\nu_A(P,g)=\inf\{j\in\mathbb N:D^j(z\mapsto P(\ell_g(z)))(0)\ne0\}$ in $\mathbb N\cup\{\infty\}$, where derivatives are full iterated Fréchet derivatives, order zero is included, and the infimum of an empty set is $\infty$. A mixed-codimension index for $H$ is a vector $r\in\mathbb N^s$ with $r_i\le d_i$ and $\sum_i r_i+\delta(H)=n$; its complementary vector is $\beta_i=d_i-r_i$, using natural subtraction, and its degree monomial is $\prod_i D_i^{r_i}$. For a finite subset $\Sigma\subseteq G$, let $\Sigma^{(j)}$ be the set of sums of exactly $j$ members of $\Sigma$, with repetition allowed, so $\Sigma^{(0)}=\{0\}$; let $N_\Sigma(H)=|\{x+H:x\in\Sigma\}|$ count distinct translated sets. Write $Z_G(P)=\{x\in G:P(\iota(x))=0\}$. For every $G$ there exists a real constant $c>0$ such that the following implication holds uniformly for every analytic datum $A$ with $\kappa_A(\{0\})=1$, every finite subset $\Sigma\subseteq G$ containing zero, every $T\in\mathbb N$, every natural degree vector $D$ with all $D_i\ge1$, and every nonzero coordinate polynomial $P$ homogeneous of multidegree $D$. Assume $\nu_A(P,g)\ge nT+1$ for every $g\in\Sigma^{(n)}$, and assume that for every connected algebraic subgroup $H$ with $A^\ast\nsubseteq H$ there exists a mixed-codimension index $r$ for $H$ such that $c\prod_iD_i^{r_i}\le(T+1)N_\Sigma(H)\mu_H(d-r)$. Then there exists a point $g\in G$ such that $g+A^\ast\subseteq Z_G(P)$. The constant is chosen before $A,\Sigma,T,D,P$, while the index in the hypothesis is chosen separately for each eligible subgroup. Here the translated analytic carrier is the entire subgroup generated by the local parametrization, not only its image on the parameter ball. There is no connectedness assumption on $G$, no explicitly assumed $n>0$, no requirement $T>0$, and no hypothesis that $P$ be nonzero somewhere on $G$: only its nonzeroness in the coordinate polynomial ring is required. The threshold still equals one when $n=0$ or $T=0$, zero-fold sums mean $\{0\}$, positive degree entries exclude zero entries, and the universal implication is vacuous for any $G$ admitting no analytic datum of the specified defined dimension one.