Support-sensitive bounds for shortest zero-sum subsequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918528437714944 |
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| author | Pop, Claudiu Ţurcaş, George C. |
| author_facet | Pop, Claudiu Ţurcaş, George C. |
| contents | 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 $MZ(S)\le n-|\supp(S)|+1$. The same bound holds for every sequence of length at least $|G|$. 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 $MZ(S)\le n-t(t-1)/2$, where $t=|\supp(S)|$, whenever $S$ has length $n$ over $C_n$ and $MZ(S)-1>n/2$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_04187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Support-sensitive bounds for shortest zero-sum subsequences Pop, Claudiu Ţurcaş, George C. Number Theory Combinatorics 11B75, 20K01, 11R27 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 $MZ(S)\le n-|\supp(S)|+1$. The same bound holds for every sequence of length at least $|G|$. 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 $MZ(S)\le n-t(t-1)/2$, where $t=|\supp(S)|$, whenever $S$ has length $n$ over $C_n$ and $MZ(S)-1>n/2$. 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. |
| title | Support-sensitive bounds for shortest zero-sum subsequences |
| topic | Number Theory Combinatorics 11B75, 20K01, 11R27 |
| url | https://arxiv.org/abs/2505.04187 |