On the Complexity of Modulo-q Arguments and the Chevalley-Warning Theorem

Abstract

We study the search problem class $\mathrm{PPA}_q$ defined as a modulo-$q$ analog of the well-known $\textit{polynomial parity argument}$ class $\mathrm{PPA}$ introduced by Papadimitriou '94. Our first result shows that this class can be characterized in terms of $\mathrm{PPA}_p$ for prime $p$. Our main result is to establish that an $\textit{explicit}$ version of a search problem associated to the Chevalley--Warning theorem is complete for $\mathrm{PPA}_p$ for prime $p$. This problem is $\textit{natural}$ in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for $\mathrm{PPA}_p$ when $p \ge 3$. Finally we discuss connections between Chevalley-Warning theorem and the well-studied $\textit{short integer solution}$ problem and survey the structural properties of $\mathrm{PPA}_q$.