Connecting Interpolation and Multiplicity Estimates in Commutative Algebraic Groups

TL;DR

This paper explores the relationship between interpolation and multiplicity estimates in commutative algebraic groups by constructing a chain of subgroups that describe the distribution of finitely generated subgroups and their zero loci.

Contribution

It introduces a novel chain of algebraic subgroups linked to obstructions in interpolation and multiplicity estimates, connecting subgroup structure to zero loci of sections.

Findings

Controlled the distribution of subgroup elements with respect to algebraic subgroups.

Characterized the zero locus of sections vanishing at subgroup points.

Unified multiplicity and interpolation estimates through subgroup chains.

Abstract

Let $G$ be a commutative algebraic group embedded in projective space and $\Gamma$ a finitely generated subgroup of $G$. From these data we construct a chain of algebraic subgroups of $G$ which is intimately related to obstructions to multiplicity or interpolation estimates. Let $\gamma_1,...,\gamma_l$ denote a family of generators of $\Gamma$ and, for any $S>1$, let $\Gamma(S)$ be the set of elements $n_1\gamma_1+..+n_l\gamma_l$ with integers $n_j$ such that $|n_j| < S$. Then this chain of subgroups controls, for large values of $S$, the distribution of $\Gamma(S)$ with respect to algebraic subgroups of $G$. As an application we essentially determine (up to multiplicative constants) the locus of common zeros of all $P \in H^0(\barG ,{\cal O}(D))$ which vanish to at least some given order at all points of $\Gamma(S)$. When $D$ is very small this result reduces to a multiplicity estimate; when $D$ is very large it is a kind of interpolation estimate.