Source corrections — dimension-zero and zero-degree obstructions

Set $K=\mathbb C$. 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 $Z_G(P)=\{x\in G:P(\iota(x))=0\}$. The theorem is the conjunction of three independent existence assertions. First, there exist $G,A,H$ such that any two points of $G$ are equal, $n=0$, $H=G$, $A^\ast=\{0\}$, $\kappa_A(H)=0$, $\deg_G(\mathbf1)=\deg_H(\mathbf1)=1$, and $\binom{0+\kappa_A(H)}{\kappa_A(H)}N_{\{0\}}(H)\deg_H(\mathbf1)\le\frac{1}{4^n n!}\deg_G(\mathbf1)$, but there does not exist any coordinate polynomial $P$ for which $\nu_A(P,h)\ge1$ for every $h\in H$ and there is an $x\in G\setminus Z_G(P)$. The displayed numerical inequality has both sides equal to one under the specified equalities. Second, there exist $G,A$, a finite subset $\Sigma\subseteq G$, and a polynomial $P$ such that $n=0$, $|\Sigma|=2$, $0\in\Sigma$, $A^\ast=\{0\}$, $P\ne0$, $P$ is homogeneous of multidegree $\mathbf1$, and $\nu_A(P,g)=\infty$ for every $g\in\Sigma^{(n)}$, yet there is no algebraic subgroup $H$ for which $g+H\subseteq Z_G(P)$ for every $g\in\Sigma$. Since $n=0$, that contact condition concerns exactly the point zero, rather than all two sample points. Third, there exist $G,A,P$ for which $s=2$, $d_0=d_1=1$, the defined analytic dimension $\kappa_A(\{0\})$ is one, $P\ne0$ is homogeneous of multidegree $D=(1,0)$, and $\nu_A(P,0)=1$, such that for every real $c>0$ and every connected algebraic subgroup $H$ with $A^\ast\nsubseteq H$ there exists a mixed-codimension index $r$ for $H$ satisfying $c\prod_{i=0}^1D_i^{r_i}\le N_{\{0\}}(H)\mu_H(d-r)$, while there is no $g\in G$ with $g+A^\ast\subseteq Z_G(P)$. The index may depend on both $c$ and $H$; $N_{\{0\}}(H)=1$, and the zero degree entry uses the convention $0^0=1$ and $0^r=0$ for positive natural $r$. Each existence assertion may use different data, analytic parametrizations always have positive parameter dimension even when their generated image is trivial, and all Hilbert-polynomial and derivative-infimum fallback conventions stated above remain part of these formulas.