The ring of integer valued polynomials on \(2 \times 2\) matrices and its integral closure (Q493781)

The authors consider the ring Int\(_Q(M_n(Z))\) of polynomials with rational coefficients, preserving the ring \(M_n(Z)\) of \(n\times n\) matrices with integer entries. They prove that the integral closure of a localization of that ring coincides with the ring Int\(_Q(R_n)\) of polynomials \(f\in Q[X]\) preserving a maximal order in a division algebra \(R_n\) of degree \(n^2\) over a local field (Proposition 2.1), and then show how to construct a regular basis for Int\(_Q(R_n)\) in the case \(n=2\) using Bhargava's method of \(p\)-orderings [\textit{M. Bhargava}, J. Reine Angew. Math. 490, 101--127 (1997; Zbl 0899.13022); J. Am. Math. Soc. 22, No. 4, 963--993 (2009; Zbl 1219.11047)]. They use this result as well as a theorem of \textit{S. Frisch} [Lect. Notes Pure Appl. Math. 241, 253--259 (2005; Zbl 1092.13027); J. Algebra 373, 414--425 (2013; Zbl 1273.13037); corrigendum ibid. 412, 282 (2014)] to describe a way of computing a regular basis for Int\(_Q(M_2(Z))\) .