E-sequences and the Stone-Weierstrass theorem (Q1937301)

Let \(V\) be a discrete valuation domain with the residue field of \(q<\infty\) elements. The authors show that if either \(q\) is odd or the maximal ideal of \(V\) is generated by \(2\), then the algebra of even integer-valued polynomials on \(V\) have a \(V\)-basis given by an explicit formula, and the same holds for the \(V\)-module of odd integer-valued polynomials on \(V\). This result is applied to the proof of the analogue of the Stone-Weierstrass theorem for even/odd continuous functions on the completion of \(V\).