Moscow--Petersburg Seminar on low-dimensional mathematics
April 22, 2005

A.Esterov (Moscow)
Newton diagrams, Poincare-Hopf indices, and resultants

The principal part and the Newton diagram of an analytical function germ
in several variables is a generalization of the notions of the leading term
and the degree of the leading term in the Tailor series of an analytical
function in one variable. Just as in one-dimensional case, discrete invariants
of an isolated singularity of an analytical function (the Milnor number etc.)
can be estimated in terms of its Newton diagram (or can be expressed in these
terms, provided that the principal part coefficients of this function are in
general position). Most of the results of this type are based on the
application of A. G. Khovanskii's toric resolutions. This results and methods
can also be generalized to complete intersection singularities.

Nevertheless, the construction of toric resolution cannot be directly applied
to more complicated singularities (such as the set of degeneration points of
a holomorphic matrix germ, a singularity of a 1-form on a singular variety
etc.). We shall generalize the construction of toric resolution to apply it
to singularities of this type. As a consequence, some discrete invariants of
such singularities (the multiplicity, the generalized Poincare-Hopf index etc.)
will be investigated in terms of Newton diagrams. New proofs and more convenient
formulations for some known results about Newton diagrams will also be given
(the Newton diagram of the generalized Gelfand-Kapranov-Zelevinski resultant etc.)

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