Algebras of constants for some extensions of derivations (Q1866458)

Let \(A= B[t]\) be the polynomial ring in one variable \(t\) over a unique factorization \(k\)-domain \(B\), where \(k\) is a field with \(\operatorname {char}k=0\). Given an arbitrary derivation \(\delta\) of \(B\), certain extensions \(d\) to \(A\) are considered and their rings of constants \(A^d\) are described, the most important ones being defined by \(d(b)= t^s \delta(b)\) and \(d(t)= (s+1)^{-1} \delta(\varphi)\) for any \(s\in \mathbb{N}\) and \(\varphi\in B\) with \(\delta(\varphi)\neq 0\). The same is done for \(A_0^{\overline{d}}\), where \(\overline{d}\) denotes the extension of \(d\) to the field of quotients \(A_0\) of \(A\). Among the presented applications are two new counterexamples to the 14th problem of Hilbert.