The ring of integer valued polynomials on 2×22×2 matrices and its integral closure

Let Mn(Z)Mn(Z) denote the ring of n×nn×n matrices with integer entries and IntQ(Mn(Z))⊆Q[x]IntQ(Mn(Z))⊆Q[x] the algebra of polynomials that preserve Mn(Z)Mn(Z), i.e. polynomials for which f(M)∈Mn(Z)f(M)∈Mn(Z) if M∈Mn(Z)M∈Mn(Z). The aim of this paper is to shed some light on this algebra by establishing two results. The first, general, result is to show that when this algebra is localized at a prime p  , its integral closure can be identified with IntQ(Rn,p)IntQ(Rn,p), the ring of polynomials in Q[x]Q[x] preserving the maximal order in a division algebra of dimension n2n2 over QpQp, the p-adic numbers. This is of interest computationally because one of the authors showed in an earlier paper that Bhargava's method of p  -ordering can be extended to construct a regular basis for algebras of this type. Thus we have a method of computing an upper bound for the size of IntQ(Mn(Z))IntQ(Mn(Z)). Our second result is a computational method for constructing a p  -local basis for IntQ(Mn(Z))IntQ(Mn(Z)) itself in the case n=2n=2 in low degrees. These results together give an estimate of the difference in size between IntQ(M2(Z))IntQ(M2(Z)) and its closure and, in particular, give constructions of polynomials in the latter but not in the former.