By Ciprian S. Borcea (auth.), Jürgen Herzog, Victor Vuletescu (eds.)

Commutative Algebra, Singularities and machine Algebra provides present tendencies in commutative algebra, algebraic combinatorics, singularity conception and desktop algebra, and highlights the interplay among those disciplines. Contributions by means of major overseas mathematicians completely talk about themes in: modules thought, integrally closed beliefs and determinantal beliefs, singularities in projective areas and Castelnuovo-Mumford regularity, Groebner and SAGBI foundation, and using the pc programs Bergman, CoCoA and SINGULAR.

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Extra info for Commutative Algebra, Singularities and Computer Algebra: Proceedings of the NATO Advanced Research Workshop on Commutative Algebra, Singularities and Computer Algebra Sinaia, Romania 17–22 September 2002

Example text

Since the modules involved arc bigraded, so is Ii. Let Ti) be the direct sum of all the components of 1i of bidegree (j ,k) as k varies. Since 1i is a finitely generated bigraded S-module, Ii) is a finitely generated and graded K[tJ-module (with respect to the standard grading of K[t]). So we may decompose it as Ii j = Fi j ID Gij where Fii is the free part and Gii is the torsion part, which, being K [tJ-graded, is a direct sum of modules of the form K[tl /( t a ) for various a > O. Denote the minimal number of generators of Fij and Gii as K[tJ-modules by iii and gii' respectively.

A monomial order satisfies the following conditions: t is a total order < r on the set M (R) which (a) I

5]. The elements of Ri arc a-homogeneous of a-degree i. We say that a vector subspace V of R is a-graded if it is generated, as a vector space, by homogeneous clements. 4. Let V be a K-suospace of R. (a) 1f mE M(V) then there exists fm E V such that in(f) = m. The polynomial fm is uniquely determined if we further require that the support of fm intersects M(V) exactly in m and that fm has leading coefficient 1. (b) M(V) is a K-b asis D/in(V). (e) The set {1m : m E M(V) ) is a K-basis D/V. (d) lfV hasfinite dimension, then dim(V) == dim(in(V».

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