Mixed Newton numbers and isolated complete intersection singularities

Abstract

Let f:(ℂn, 0) → (ℂp, 0) be a complete intersection with an isolated singularity at the origin. We give a lower bound for the Milnor number of f in terms of the mixed multiplicities of a set of monomial ideals attached to the Newton polyhedra of the component functions of f. The Milnor number of f equals the bound that we give when f satisfies a condition that we define and that extends the notion of Newton non‐degenerate function studied by Kouchnirenko. Our techniques are based on the notion of integral closure of submodules and its relation with Buchsbaum–Rim multiplicity and mixed multiplicities of a set of ideals.

Torical modification of Newton non-degenerate ideals

“…Newton non degenerate singularities of hypersurfaces and of complete intersections have been widely studied (See for example [BrNe, Kou] and [Kho1,Oka1,Biv,SaZu]). A good resolution of a non degenerate singularity may be constructed from the dual fan of the Newton boundaries.…”

The sequence of mixed Łojasiewicz exponents associated to pairs of monomial ideals

“…Let us consider the ideal I of O 2 given by I = x 4 + y 2 , x 5 . We observe that e(I) = 10 = L (1) 0 (I)L (2) 0 (I). The set of vertices of Γ + (I) is {(4, 0), (0, 2)}.…”

“…. , g n is a non-degenerate sequence and h i is a generic C-linear combination of a fixed generating system of I i , we can suppose, by [2,Lemma 5.5], that h i , g i+1 , . .…”

Local tropicalizations of splice type surface singularities

“…A slightly different notion of Newton non-degenerate ideals was introduced by Saia in [32] for ideals of finite codimension in Ô. For a comparison with Khovanskii's approach we refer the reader to Bivià-Ausina's work [3,Lemma 6.8]. For a general perspective on Newton non-degenerate complete intersections, the reader can consult Oka's book [23].…”