Advances and improvements in the theory of standard bases and syzygies
The aim of this article is to describe recent advances and improvements on the tangent cone algorithm of \textit{T. Mora}. This tangent cone algorithm is itself a variant of B. Buchberger's celebrated algorithm for constructing a Gröbner basis of an ideal in a polynomial ring over a field. In the same manner as the knowledge of a Gröbner basis allows the computation of numerous invariants of the coordinate ring of a projective algebraic variety, a standard basis (computed by the tangent cone algorithm) does so for invariants of the local ring of an algebraic variety at a given point. In this paper we describe a generalization which includes Buchberger's and Mora's algorithm as special cases. That is, we prove -- with an appropriate definition of ecart -- that Mora's algorithm terminates for any ordering on the monomials of \(K[x_1, \dots, x_n]\), which is compatible with the natural semigroup structure (a fact which was found independently by Gräbe), in particular, the variables may have as well negative, positive or zero weights. More or less all algorithms using Gröbner bases (such as computation of syzygies, ideal theoretic operations, etc.) are now available in this general context. Our generalization provides also an easy manner to implement standard bases for modules over the Weyl algebra and for \({\mathcal D}\)-modules. The authors give also a proof of (a sharpened version) of Hilbert's syzygy theorem for localizations of polynomial rings over a field with respect to a general semigroup ordering. This proof, which generalizes Schreyer's proof for polynomial rings, uses just the existence of standard bases and is quite elementary. Using a deformation connecting the module of leading terms with the module itself, it is proven that standard bases can be used to compute numerical invariants of a module like the dimension, and under certain natural restrictions on the ordering in the local case, the multiplicity and Hilbert function. Computer experiments with an implementation of the described algorithms in the system SINGULAR have prompted a partial positive answer to Zariski's multiplicity conjecture which is also contained in the paper.
invariants of the local ring of an algebraic variety
scientific article; zbMATH DE number 937062
scientific article