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The paper is concerned with an effective formula for the Euler characteristic of the link of a weighted homogeneous mapping F : R → R with an isolated singularity. The formula is based on Szafraniec’s method for calculating the Euler characteristic of a real algebraic manifold (as the signature of an appropriate bilinear form). It is shown by examples that in the case of a weighted homogeneous mapping it is possible to make the computer calculations of the Euler characteristics much more effective. Let F : R → R, where n > k + 1, be a polynomial mapping and let L = {x ∈ Sn−1 | F (x) = 0}. Let χ(L) denote the Euler characteristic of L. There is an algebraic formula, due to Szafraniec [7], which expresses χ(L) in terms of the signature of an appropriate bilinear form (provided that L is smooth). The aim of this paper is to show that in the weighted homogeneous case the general formula for the Euler characteristic of L can be replaced by a simpler one. It makes practical (computer) calculations of the Euler characteristic more effective. Actually, in some cases the general method is ineffective because of huge computations. The basic idea is to consider the intersection of F−1(0) with the hyperplanes {xn = 0} and {xn = 1}. We prove that the Euler characteristic of L is determined by the behavior of F on these spaces, so the formula for χ(L) becomes simpler (we consider only the case when the manifold L is of even dimension, for otherwise its Euler characteristic is zero). We also consider the simpler invariant χ(L)/2 (mod 2); we give a simple formula which requires only calculating the dimension of an appropriate algebra (instead of computing a signature). There is a computer program by Andrzej Łęcki to compute the dimensions of quotient algebras and signatures of bilinear forms on these algebras; 2000 Mathematics Subject Classification: 14P05, 57R70.

On the Euler characteristic of the link of a weighted homogeneous mapping