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

Authors Mika Göös, Pritish Kamath, Katerina Sotiraki, Manolis Zampetakis

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Author Details

Mika Göös
  • Stanford University, CA, USA
Pritish Kamath
  • Toyota Technological Institute at Chicago, IL, USA
Katerina Sotiraki
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Manolis Zampetakis
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Göös, Mika: Work done while at IAS. Supported by NSF grant CCF-1412958.
  • Kamath, Pritish: Work done while at MIT. Supported in part by NSF Awards CCF-1733808 and IIS-1741137 and MIT-IBM Watson AI Lab and Research Collaboration Agreement No. W1771646.
  • Sotiraki, Katerina: Supported in parts by NSF/BSF grant #1350619, an MIT-IBM grant, and a DARPA Young Faculty Award, MIT Lincoln Laboratories and Analog Devices.
  • Zampetakis, Manolis: Supported by a Google PhD Fellowship.

Acknowledgements

We thank Christos Papadimitriou, Robert Robere, Dmitry Sokolov and Noah Stephens-Davidowitz for helpful discussions. We also thank anonymous referees for valuable suggestions.

Cite AsGet BibTex

Mika Göös, Pritish Kamath, Katerina Sotiraki, and Manolis Zampetakis. On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 19:1-19:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.19

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • Total NP Search Problems
  • Modulo-q arguments
  • Chevalley - Warning Theorem

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