$Q_p$-weighted zero-sum constants
Krishnendu Paul, Shameek Paul
TL;DR
This work determines weighted zero-sum constants over prime fields $\ Z_p$ with quadratic-residue weights. Using Chevalley–Warning arguments and structural properties of quadratic residues, the authors establish exact values for $E_{Q_p,\mathbf 1}$, $C_{Q_p,\mathbf 1}$, and $D_{Q_p,\mathbf 1}$, plus precise results for $(Q_p,B)$-weighted constants, including the special case $p=5$. Key findings are $E_{Q_p,\mathbf 1}=p+2$ for $p\neq5$ and $=9$ for $p=5$, $C_{Q_p,\mathbf 1}=6$ when $p\equiv1\pmod{4}$ and $=9$ when $p\equiv3\pmod{4}$, and $D_{Q_p,\mathbf 1}=4$ or $5$ with a conjectured $4$ for $p\equiv1\pmod{4}$; for $B\subseteq\mathbb Z_p$ (with $p\neq5$) one gets $E_{Q_p,B}=p+2$ and explicit $C_{Q_p,B}$ and $D_{Q_p,B}$ depending on $p\bmod 4$ and $B$'s relation to $Q_p$. The results advance weighted zero-sum theory in finite fields and lay groundwork for further generalizations to $(Q_p,B)$-weighted setups.
Abstract
A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_p$ has a $(Q_p,\mathbf 1)$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf 1}$ and the related constants $C_{Q_p,\mathbf 1}$ and $D_{Q_p,\mathbf 1}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$.