The Łojasiewicz exponent of semiquasihomogeneous singularities

brzostowski, szymon;

Let f : (Cn, 0) → (C, 0) be a semiquasihomogeneous function. We give a formula for the local Łojasiewicz exponent L0(f) of f, in terms of weights of f. In particular, in the case of a quasihomogeneous (QH) isolated singularity f, we generalize a formula for L0(f) of Krasiński,
Oleksik and Płoski from 3 to n dimensions. This was previously announced in the paper [19] of Tan, Yau and Zuo [Łojasiewicz inequality for weighted homogeneous polynomial with isolated
singularity, Proc. Amer. Math. Soc. 138 (2010) 3975–3984], but as a matter of fact it was not proved correctly there, as noted by the AMS reviewer Tadeusz Krasiński. As a consequence of our result, we obtain that the Łojasiewicz exponent is invariant in topologically trivial families of singularities coming from a QH germ. This is an affirmative
partial answer to Teissier’s conjecture.