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.

This is an extended abstract on the equivalence, in the context of commutative formal power series, of two confluence notions arising in topological rewriting theory, namely topological confluence and infinitary confluence. The topological rewriting relation studied is, in general, not quite the same as the one computer scientists investigate for infinitary λ/Σ-terms: we provide an example where they differ for formal power series and we formulate the so-called 'chains conjecture' that ensures that, if we have finitely many rewrite rules, the two relations coincide.

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.

Introducing notions from topological rewriting, we present the result from our latest submission proving the equivalence in the context of commutative formal power series of two new notions of confluence: topological confluence and infinitary confluence. We mention an alternative proof of the this result and a previous theorem that we require in terms of solely topological rewriting methods. Finally, we enunciate the 'chains conjecture' that aims in particular to prove that the topological rewriting relation of our research correspond to the relation investigated by computer scientists for infinitary λ/Σ-terms.

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 lambda-calculus. 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.

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 I-adic 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.

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, non-commutative 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 n-chains. 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 so-called overlap property.

This is the working notes on my research in collaboration with Cyrille Chenavier and Thomas Cluzeau on the topic of topological rewriting theory. It is an extension of classical rewriting theory that takes more interest in the cases of non-terminating rewriting systems as well as rewriting systems in which the classical notion of confluence is too restrictive to depict the phenomena that occur. In these notes, we start by reminding some notions from topology before presenting the basic notions and results of topological rewriting theory. Finally, we give the concrete example of rewriting on the algebra of commutative formal power series in several variables.

Anick 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 non-commutative Groebner bases.