1. On power sums of matrices over a finite commutative ring
- Author
-
Pedro Fortuny Ayuso, Ignacio F. Rúa, José Maria Grau, José María Grau Ribas, and Antonio M. Oller-Marcen
- Subjects
Discrete mathematics ,Conjecture ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Commutative ring ,01 natural sciences ,Matrix ring ,Power (physics) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,0103 physical sciences ,Idempotence ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Computer Science::General Literature ,010307 mathematical physics ,0101 mathematics ,Element (category theory) ,Value (mathematics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].
- Published
- 2017