Erhard Aichinger,
"Solving systems of equations in supernilpotent algebras"
, Serie arXiv.org, Nummer arXiv:1901.07862v1, 1-2019, ISSN: 2331-8422
Original Titel:
Solving systems of equations in supernilpotent algebras
Sprache des Titels:
Englisch
Original Kurzfassung:
Recently, M. Kompatscher proved that for each finite supernilpotent algebra $\mathbf{A}$ 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 $\mathbf{A}$, 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 $\mathbf{A}$, there is $c \in \mathbb{N}$ 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.
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