Solving Systems of Equations in Supernilpotent Algebras
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.
Supernilpotent algebras
polynomial equations
polynomial mappings
circuit satisfiability
Mathematics of computing~Combinatorial algorithms
Theory of computation~Complexity classes
72:1-72:9
Regular Paper
Supported by the Austrian Science Fund FWF P29931: Clonoids: a unifying approach to equational logic and clones
A preliminary version is available on https://arxiv.org/abs/1901.07862.
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.
Erhard
Aichinger
Erhard Aichinger
Institute for Algebra, Johannes Kepler University Linz, Linz, Austria
https://www.jku.at/institut-fuer-algebra/
https://orcid.org/0000-0001-8998-4138
10.4230/LIPIcs.MFCS.2019.72
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Erhard Aichinger
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