Deterministically Computing Reduction Numbers of Polynomial Ideals
نویسندگان
چکیده
We present two approaches to compute the (absolute) reduction number of a polynomial ideal. The first one puts the ideal into a position such that the reduction number of its leading ideal can be easily read off the minimal generators and then uses linear algebra to determine the reduction number of the ideal itself. The second method computes via a Gröbner system not only the absolute reduction number but the set of all reduction numbers of the given ideal and thus in particular also its big reduction number.
منابع مشابه
Topics on the Ratliff-Rush Closure of an Ideal
Introduction Let be a Noetherian ring with unity and be a regular ideal of , that is, contains a nonzerodivisor. Let . Then . The :union: of this family, , is an interesting ideal first studied by Ratliff and Rush in [15]. The Ratliff-Rush closure of is defined by . A regular ideal for which is called Ratliff-Rush ideal. The present paper, reviews some of the known prop...
متن کاملLascoux-style Resolutions and the Betti Numbers of Matching and Chessboard Complexes
This paper generalizes work of Lascoux and Jo zeeak-Pragacz-Weyman computing the characteristic zero Betti numbers in minimal free resolutions of ideals generated by 2 2 minors of generic matrices and generic symmetric matrices, respectively. In the case of 2 2 minors, the quotients of certain polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we co...
متن کاملNumerical Hilbert functions for Macaulay2
The NumericalHilbert package for Macaulay2 includes algorithms for computing local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. These techniques are numerically stable, and can be used with floating point arithmetic over the complex numbers. They provide a viable alternative in this setting to purely symbolic methods such as standard bases. ...
متن کاملIdeals containing the squares of the variables
We study the Betti numbers of graded ideals containing the squares of the variables, in a polynomial ring. We prove the lex-plus-powers conjecture for such ideals.
متن کاملDimension-Dependent Upper Bounds for Gröbner Bases
We improve certain degree bounds for Gröbner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension(and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees ...
متن کامل