Solving Systems of Equations in Supernilpotent Algebras

Author Erhard Aichinger

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Erhard Aichinger
  • Institute for Algebra, Johannes Kepler University Linz, Linz, Austria


The author thanks G. Horváth and M. Kompatscher for dicussions on solving equations over nilpotent algebras. Several of these discussions took place during a workshop organized by P. Aglianò at the University of Siena in June 2018. The author also thanks A. Földvári, C. Szabó, M. Kompatscher, and S. Kreinecker for their comments on preliminary versions of the manuscript, and the anonymous referees for several useful suggestions.

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Erhard Aichinger. Solving Systems of Equations in Supernilpotent Algebras. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 72:1-72:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Recently, M. Kompatscher proved that for each finite supernilpotent algebra A in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let mu be the maximal arity of the fundamental operations of A, and let d := |A|^{log_2 mu + log_2 |A| + 1}. Applying a method that G. Károlyi and C. Szabó had used to solve equations over finite nilpotent rings, we show that for A, there is c in N such that a solution of every system of s equations in n variables can be found by testing at most c n^{sd} (instead of all |A|^n possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Complexity classes
  • Supernilpotent algebras
  • polynomial equations
  • polynomial mappings
  • circuit satisfiability


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