# Algebra Seminar

Title: Some additive combinatorial problems over finite fields

Abstract: Let $G$ be a finite abelian group, written additively. The Davenport constant $D(G)$ is the smallest positive number $s$ such that for any set $\{g_1, g_2, \ldots, g_s\}$ of $s$ elements in $G$, with repetition allowed, there exists a subset $\{g_{i_1}, g_{i_2}, \ldots, g_{i_t}\}$ such that $g_{i_1} + g_{i_2} + \cdots +g_{i_t} =0$. The plus-minus Davenport constant, $D_{\pm}(G)$, is defined similarly but instead we only require that $g_{i_1} \pm g_{i_2} \pm \cdots \pm g_{i_t} =0$. In this talk, we provide current results for $D_\pm(G)$ for general finite abelian group, $G$. Then we present current results for groups such as $C_3^r \oplus C_5^s$ and $C_2^q \oplus C_3^r$. In working on the plus-minus Davenport constant, we came upon an interesting problem with recantular matrices with entires in finite fields. For a positive integer $m$, let $h_q(m)$ denote the largest integer $n$ such that there exists an $m \times n$ matrix $M$ with entries in $\FF_q$ where every $m \times m$ submatrix $W$ of $M$ has rank $m$. We will present our current results on the constant $h_q(m)$.