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Connecting Interpolation and Multiplicity Estimates in Commutative Algebraic Groups

Let $G$ be a commutative algebraic group embedded in projective space and $Γ$ 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 $γ_1,...,γ_l$ denote a family of generators of $Γ$ and, for any $S>1$, let $Γ(S)$ be the set of elements $n_1γ_1+..+n_lγ_l$ with integers $n_j$ such that $|n_j| < S$. Then this chain of subgroups controls, for large values of $S$, the distribution of $Γ(S)$ 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 $Γ(S)$. When $D$ is very small this result reduces to a multiplicity estimate; when $D$ is very large it is a kind of interpolation estimate.