The complexity of the equation solvability problem over semipattern groups

Földvári, Attila

doi:10.1142/s0218196717500126

Cited by 9 publications

(4 citation statements)

References 13 publications

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“…[KS15]). In [Föl17,Föl18], A. Földvári provides polynomial time algorithms for solving equations over finite nilpotent groups and rings relying on the structure theory of these algebras. In this paper, we extend the method developed in [KS18] from finite nilpotent rings to arbitrary finite supernilpotent algebras in congruence modular varieties.…”

Section: Introductionmentioning

confidence: 99%

Solving systems of equations in supernilpotent algebras

Aichinger

2019

Preprint

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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 µ be the maximal arity of the fundamental operations of A, and letApplying 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 ∈ N such that a solution of every system of s equations in n variables can be found by testing at most cn sd (instead of all |A| n possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.2010 Mathematics Subject Classification. 08A40 (68Q25).

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Section: Introductionmentioning

confidence: 99%

Solving systems of equations in supernilpotent algebras

Aichinger

2019

Preprint

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“…In particular nilpotency does not demark the border between problems in P and NP-complete: By [15] the equation solvability over the non-nilpotent group A 4 is in P but its extension by the commutator [·, ·] has an NP-complete equation solvability problem. More general, metaabelian groups [10] and semipattern groups [4] induce equation solvability problems that are in P, while not necessarily being nilpotent.…”

Section: Introductionmentioning

confidence: 99%

The equation solvability problem over supernilpotent algebras with Mal’cev term

Kompatscher¹

2018

Int. J. Algebra Comput.

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In 2011 Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal'cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal'cev term that have coNP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth's question is negative in general (assuming P =NP).

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“…, Y β ] be polynomials in expanded form. We then say (following the notation from [4]) that f | Fq,S1,...,S β = g| Fq,S1,...,S β is solvable if there exist field elements s 1 , . .…”

Section: Preliminariesmentioning

confidence: 99%

Solving a fixed number of equations over finite groups

Nuspl

2021

Algebra Univers.

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We investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed $$\mathrm {NP}$$ NP -completeness of this problem for non-Abelian groups. We show that the problem can become tractable for some non-Abelian groups if we fix the number of equations. Recently, Földvári and Horváth showed that a single equation over groups which are semidirect products of a p-group with an Abelian group can be solved in polynomial time. We generalize this result and show that the same is true for systems with a fixed number of equations. This shows that for all groups for which the complexity of solving one equation has been proved to be in $$\mathrm {P}$$ P so far, solving a fixed number of equations is also in $$\mathrm {P}$$ P . Using the collecting procedure presented by Horváth and Szabó in 2006, we furthermore present a faster algorithm to solve systems of equations over groups of order pq.

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The complexity of the equation solvability problem over semipattern groups

Cited by 9 publications

References 13 publications

Solving systems of equations in supernilpotent algebras

Solving systems of equations in supernilpotent algebras

The equation solvability problem over supernilpotent algebras with Mal’cev term

Solving a fixed number of equations over finite groups

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