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

Mika Göös, Pritish Kamath, Katerina Sotiraki, Manolis Zampetakis

Abstract:

We study the search problem class PPAq defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou '94. Our first result shows that this class can be characterized in terms of PPAp 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 PPAp 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 PPAp 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 PPAq.