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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 thatthis 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 asearch problem associated to the Chevalley--Warning theorem is complete for$\mathrm{PPA}_p$ for prime $p$. This problem is $\textit{natural}$ in that itdoes not explicitly involve circuits as part of the input. It is the first suchcomplete problem for $\mathrm{PPA}_p$ when $p \ge 3$.  Finally we discuss connections between Chevalley-Warning theorem and thewell-studied $\textit{short integer solution}$ problem and survey thestructural properties of $\mathrm{PPA}_q$.

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