## Institute for Mathematical Sciences

## Preprint ims95-11

** Roberto Silvotti**
* On a conjecture of Varchenko*

Abstract: Varchenko conjectured that, under certain genericity conditions, the number of critical points of a product $\phi$
of powers of linear functions on $\Bbb C^n$ should be given by
the Euler characteristic of the complement of the divisor of
$\phi$ (i.e., a union of hyperplanes). In this note two
independent proofs are given of a direct generalization of
Varchenko's conjecture to the case of a generalized meromorphic
function on an algebraic manifold whose divisor can be any
(generally singular) hypersurface. The first proof uses
characteristic classes and a formula of Gauss--Bonnet type for
affine algebraic varieties. The second proof uses Morse theory.

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