Bases of integer-valued polynomials (Q2785941)

Let \(R\) be a Dedekind domain with quotient field \(K\). The ring of integer-valued polynomials on \(R\) is \(\text{Int} (R)= \{f\in K [X]\mid f(R) \subseteq R\}\). Let \({\mathcal I}_n\) be the fractional ideal of \(R\) consisting of 0 together with the leading coefficients of the elements \(f\in \text{Int} (R)\) of degree \(m\leq n\). It is known that there exist monic polynomials \(f_0\), \(f_1, \dots\), such that \(f_i\) has degree \(i\) and \(\text{Int} (R)= \bigoplus^\infty_{n=1} {\mathcal I}_n f_n\). Then \(\text{Int} (R)\) has a regular basis, that is a free basis \(g_0\), \(g_1, \dots\), with \(\deg (g_i) =i\) for each \(i\), if and only if each \({\mathcal I}_n\) is principal. (For example, \(\text{Int} (\mathbb{Z} [\sqrt {-5}])\) does not have a regular basis.) However, since \(\text{Int} (R)\) is a projective module of infinite rank, it is a free \(R\)-module, even if the ideals \({\mathcal I}_n\) are not principal. NEWLINENEWLINENEWLINEIn the paper under review the author shows how to construct a basis of \(\text{Int} (\mathbb{Z} [\sqrt{-5}])\) which is close to being a regular basis.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].