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

Abstract

We study the search problem class PPA q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA p for prime p.Our main result is to establish that an explicit version of a search problem associated to the Chevalley-Warning theorem is complete for PPA p for prime p. This problem is natural in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for PPA p when p ≥ 3.Finally we discuss connections between Chevalley-Warning theorem and the well-studied short integer solution problem and survey the structural properties of PPA q .