2022/12/29 by Eric Swartz, Swartz, Eric, Nicholas J. Werner +1 · 1 citation
Computer Science · Mathematics · #05C25 #15A15 #15B33 #16S50 #Advanced Topics in Algebra #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2212.14460
openalex publication_date 2022/12/29 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28
Let F be a field and Mn(F) the ring of n × n matrices over F. Given a subset S of Mn(F), the null ideal of S is the set of all polynomials f with coefficients from Mn(F) such that f(A) = 0 for all A ∈ S. We say that S is core if the null ideal of S is a two-sided ideal of the polynomial ring Mn(F)[x]. We study sufficient conditions under which S is core in the case where S consists of 3 × 3 matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if F is finite with q elements and |S| \geqslant q3-q2+1, then S is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.