Vanishing of Multiple Zeta Values over ${\mathbb{F}}_q[t]$ at Negative Integers

Abstract

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements. In this paper, we study the vanishing behavior of multizeta values over ${\mathbb{F}}_q[t]$ at negative integers. These values are analogs of the classical multizeta values. At negative integers, they are series of products of power sums $S_d(k)$ which are polynomials in $t$. By studying the $t$-valuation of $S_d(s)$ for $s<0$, we show that multizeta values at negative integers vanish only at trivial zeros. The proof is inspired by the idea of Sheats in the proof of a statement of "greedy element" by Carlitz.

19 pages