Newton filtrations, graded algebras and codimension of non-degenerate ideals

Abstract

We investigate a generalization of the method introduced by Kouchnirenko to compute the codimension (colength) of an ideal under a certain non-degeneracy condition on a given system of generators of I. We also discuss Newton non-degenerate ideals and give characterizations using the notion of reductions and Newton polyhedra of ideals.

Mixed Łojasiewicz exponents and log canonical thresholds of ideals

“…Consequently f t is a semi-weighted homogeneous function, for all t, by [9,Theorem 3.3] (see also [16]). Then, by [7,Corollary 4.7], we obtain…”

“…We refer to ν Γ as the Newton filtration induced by Γ + (see also [8,9,26] for the case where Γ + is convenient). If J is an ideal of O n , then we define ν Γ (J) = min{ν Γ (g) : g ∈ J}.…”

Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals

“…This notion motivated in turn the definition of Newton non-degenerate ideal (see [8,10] or [32]). Let I be an ideal of O n and let g 1 , .…”

“…As a consequence we have θ(I 1 , I 2 , I 3 ) = Example 4.6 Let us consider the ideals I 1 = x 5 , x 2 y 2 , y 5 and I 2 = x 3 y 3 of O 2 . Any element g = (g 1 , g 2 ) ∈ R 0 (I 1 , I 2 ) verifies that g is non-degenerate with respect to the Newton filtration in O 2 defined by + (I 1 ) (see the details about this definition in [10]). …”

DEFORMATIONS WITH CONSTANT MILNOR NUMBER AND MULTIPLICITY OF COMPLEX HYPERSURFACES The first named author is partially supported by CNPq-Grant 30055692-6.

“…In order to generalize these results to a bigger class of germs that includes the Newton non-degenerate germs and also the class of semi-weighted homogeneous germs, we apply the results of Bivia-Fukui-Saia, given in [2], to give a necessary and sufficient condition for the µ-constancy of families defined by germs which are non-degenerate on some Newton polyhedron. We repeat the basic results for this definition.…”