Research
Preprints

Topological closure of formal power series ideals and application to topological rewriting theory
Author(s): Cyrille CHENAVIER. Thomas CLUZEAU. Adya MUSSONLEYMARIE.
20240208
arXiv (submitted to Journal of Symbolic Computation)
DOI: https://doi.org/10.48550/arXiv.2402.05511We 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 to topological rewriting theory. In this context, 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 that 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 confluence are equivalent.
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Conference talks

Confluence of topological rewriting systems
Author(s): Adya MUSSONLEYMARIE.
20240326
FELIM  Functional Equations in LIMoges
DOI: #Computations in multivariate formal power series make use of the concept of standard bases of ideals in an analogous manner as Gröbner bases are used for ideals in multivariate polynomials rings to algorithmically solve problems such as ideal membership, amongst other problems. Extending the classical rewriting theory to a topological setting, such as the complete metric space of formal power series, it has been shown that the property of being a standard basis is characterised by a certain confluence property of the rewriting relation. Using the fact that ideals of formal power series are topologically closed, we proved that this confluence property is equivalent to a generally stronger confluence property. This latter property is studied by computer scientists in the context of infinitary term rewriting and infinitary lambdacalculus. This result falls into the preliminary research towards the ultimate goal of applying topological rewriting theory to the study of formal solutions of partial differential equations.
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Seminars

Topological closure of ideals of commutative formal power series and applications
Author(s): Adya MUSSONLEYMARIE.
20240606
XLIM  Séminaire Calcul Formel
DOI: #Motivated by applications to the new theory of topological rewriting and especially in the context of rewriting on formal power series, one is bound to ask themselves about the topological properties of the objects involved; for instance, algebraic ideals in the ring of the formal power series. From a general result of Zariski and Samuel, it is known that ideals of commutative formal power series are topologically closed for the Iadic topology induced by the ideal generated by the indeterminates. By means of rewriting methods using standard bases (analogous to Gröbner bases), we present in that talk a constructive proof of this property. Finally, we mention the implications this result has in the equivalence of different confluence properties and in the characterisations of standard bases.
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Theses

Masters thesis: Homological algebra (Anick resolution, Koszul complex and noncommutative Gröbner bases)
Author(s): Adya MUSSONLEYMARIE.
202309
Unpublished
DOI: #This thesis provides all necessary introductory knowledge for a graduate student in mathematics to grasp the basics of homological algebra as well as the objects studied in the contribution part. This includes an introduction to module theory, tensor products over arbitrary rings, associative unitary algebras, noncommutative Gröbner bases, category theory and homological algebra (in particular the language of the Tor and Ext functors). Then, the Anick resolution for augmented algebras is described giving three different but equivalent definitions of the central ingredient: the nchains. After giving an overview of the Koszul duality for homogeneous algebras, we explain the Koszul complex in that context. Finally, the contribution of this work is a proof of equality between the Anick resolution and the Koszul complex in the context of homogeneous monomial algebras satisfying the socalled overlap property.
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Notes

On Anick resolution: from the original setting to the language of noncommutative Groebner bases
Author(s): Adya MUSSONLEYMARIE.
202307
arXiv
DOI: https://doi.org/10.48550/arXiv.2307.06144Anick introduced a resolution, that now bears his name, of a field using an augmented algebra over that field. We present here what one could call a dictionary between Anick's original paper and the other resources on the matter, most of which use the language of noncommutative Groebner bases.
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