🤖 AI Summary
This paper addresses computational challenges concerning non-finitely generated ideals in complete local equicharacteristic Noetherian rings—particularly formal power series rings—where classical Gröbner basis theory fails.
Method: It introduces, for the first time, purely topological rewriting tools into standard basis theory, integrating generalized confluence analysis, standard basis algorithms over formal power series rings, and algebraic techniques including the Cohen Structure Theorem.
Contribution/Results: The work rigorously reconstructs and proves a generalized confluence characterization of standard bases; establishes, for the first time, logical equivalence between two distinct notions of generalized confluence in formal power series rings; and thereby constructs a unified computational framework applicable to non-finitely generated ideals. This significantly extends both the scope and theoretical depth of Gröbner-type theories, bridging standard basis theory with topological rewriting theory through a profound structural analogy.
📝 Abstract
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.