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Adya Musson-Leymarie

Publications and source records attributed to Adya Musson-Leymarie.

4 recordsLinked to original sources
arXiv2026

Schreyer resolution of modules over formal power series

Standard bases of modules over algebras of formal power series play the same role as Gröbner bases of modules over polynomial algebras. In this article, we first prove the analogue of the diamond lemma for modules over formal power series; it characterises standard bases in terms of unique remainders and standard representations. Then, using standard representations, we provide a method to construct a standard basis of the module of syzygies of a standard basis. This construction can be applied inductively to obtain a free resolution, similar to the Schreyer resolution, for finitely presented modules over formal power series.

math.AC↗
arXiv2025

Computing in complete local equicharacteristic Noetherian rings via topological rewriting on commutative formal power series

In commutative algebra, the theory of Gröbner bases enables one to compute in any finitely generated algebra over a given computable field. For non-finitely generated algebras however, other methods have to be pursued. For instance, it follows from the Cohen structure theorem that standard bases of formal power series ideals offer a similar prospect but for complete local equicharacteristic rings whose residue field is computable. Using the language of rewriting theory, one can characterise Gröbner bases in terms of confluence of the induced rewriting system. It has been shown, so far via purely algebraic tools, that an analogous characterisation holds for standard bases with a generalised notion of confluence. Subsequently, that result is utilised to prove that two generalised confluence properties, where one is actually in general strictly stronger than the other, are actually equivalent in the context of formal power series. In the present paper, we propose alternative proofs making use of tools purely from the new theory of topological rewriting to recover both the characterisation of standard bases and the equivalence between generalised confluence properties. The objective is to extend the analogy between Gröbner basis theory together with classical algebraic rewriting theory and standard basis theory with topological rewriting theory.

math.AC↗
arXiv2024

Topological closure of formal power series ideals and application to topological rewriting theory

We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.

math.AC↗