eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-08-20
72:1
72:9
10.4230/LIPIcs.MFCS.2019.72
article
Solving Systems of Equations in Supernilpotent Algebras
Aichinger, Erhard
1
https://orcid.org/0000-0001-8998-4138
Institute for Algebra, Johannes Kepler University Linz, Linz, Austria
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol138-mfcs2019/LIPIcs.MFCS.2019.72/LIPIcs.MFCS.2019.72.pdf
Supernilpotent algebras
polynomial equations
polynomial mappings
circuit satisfiability