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Assume that the coefficients of a polynomial in a complex variable areLaurent polynomials in some complex parameters. The parameter space (a complextorus) splits into strata corresponding to different combinations ofcoincidence of the roots of the polynomial. For generic Laurent polynomialswith fixed Newton polyhedra the Euler characteristics of these strata are alsofixed. We provide explicit formulae for the Euler characteristics of the stratain terms of the polyhedra of the Laurent polynomials in the cases of degrees 2and 3. We also obtain some corollaries in combinatorial geometry, which followsfrom two different ways of computing the Euler characteristic of thebifurcation set for a reduced polynomial of degree 2.

Euler characteristic of the bifurcation set for a polynomial of degree 2 or 3