Support-sensitive bounds for shortest zero-sum subsequences

For a sequence $S$ over a finite abelian group, let $MZ(S)$ denote the length of the shortest nonempty zero-sum subsequence of $S$. We prove that if $G$ is finite abelian of order $n$ and $S$ has length $n$, then $MZ(S)\le n-\mathrm{\mid }\text{supp}(S)\mathrm{\mid }+1$. The same bound holds for every sequence of length at least $\mathrm{\mid }G\mathrm{\mid }$. In cyclic groups we combine this elementary support bound with the Savchev–Chen structure theorem for long zero-sumfree sequences and obtain the sharper estimate $MZ(S)\le n-t(t-1)\mathrm{/}2$, where $t=\mathrm{\mid }\text{supp}(S)\mathrm{\mid }$, whenever $S$ has length $n$ over $C_n$ and $MZ(S)-1>n\mathrm{/}2$. As a consequence, every length-$n$ sequence over $C_n$ with support size $3$ has a zero-sum subsequence of length at most $n-3$, and this is sharp for $n\ge 5$. We also give an arithmetic application to products of prime ideals in a number field, phrased in the standard class-group and block-monoid setting and a corresponding cyclic class-group sharpening.