On topological invariants mod 2 of weighted homogeneous polynomials (Q1913470)

Let \(f:\mathbb{R}^n \to\mathbb{R}\) be a weighted homogeneous polynomial with \(df (0) =0\). Let \(L= \{x\in S^{n-1} \mid\) \(f(x) =0\}\). It is a well-known fact that the Euler characteristic \(\chi(L)\) is an even number. \textit{Z. Szafraniec} [Glasg. Math. J. 33, No. 3, 241-245 (1991; Zbl 0746.14026)] gave an expression for \(\chi(L)\) in terms of the local topological degrees of two explicitly defined maps. These degrees may be computed algebraically by the Eisenbud-Levine formula, i.e. by calculating signatures of certain bilinear forms. Unfortunately, practical methods of calculating these signatures are not easy. In this paper we shall show how to calculate the simpler topological invariant \(\chi(L)/2 \bmod 2\) using simpler algebraic methods. Our approach requires only computing dimensions of appropriate local algebras. It is proper to add that we do not assume that \(f\) has an isolated singularity (hence \(L\) does not have to be smooth). In the case of homogeneous polynomials, similar work has been done by \textit{Z. Szafraniec} [Bull. Pol. Acad. Sci., Math. 37, No. 1-6, 103-107 (1989; Zbl 0766.57016)].