A normal form algorithm for modules over \(k[x,y]/\langle xy \rangle\) (Q1815025)

The authors give an algorithm to classify finitely generated modules over the ring \(R:=k[x,y]/\langle xy \rangle ,\) where \(k\) is any field. This algorithm generalizes the Smith normal form algorithm for a matrix over a polynomial ring in one variable. The authors study is motivated by the work of \textit{L. S. Levy} [J. Algebra 71, 62-114 (1981; Zbl 0508.16009) and 93, 1-116 (1985; Zbl 0564.13010)]. They derive an algorithmic version of Levy's results for \(R\). The following application to a problem in linear algebra is given: Compute the normal form of a pair of mutually annihilating \(n \times n\) matrices over \(k\) (i.e. \(C, D \in k^{n\times n}\) verifying \(CD=DC=0\)), under the action of the general linear group \((C,D) \mapsto (U^{-1} C U , U^{-1} D U)\), \(U\in GL(n,k)\). In particular, when \(D=0\), the rational form of any square matrix \(C\) can be read off the output of the proposed algorithm.