2014/05/20 by Szymon Brzostowski, Brzostowski, Szymon
Computer Science · Mathematics · #32S05 #58K05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1405.5179
openalex publication_date 2014/05/20 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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 isolated singularity f, we generalize a formula for L0(f) of Krasiński, Oleksik and Płoski ([KOP09]) from 3 to n dimensions. This was previously announced in [TYZ10], but as a matter of fact it has not been proved correctly there, as noticed by the AMS reviewer T. Krasiński. As a consequence of our result, we get that the Łojasiewicz exponent is invariant in topologically trivial families of singularities coming from a quasihomogeneous germ. This is an affirmative partial answer to Teissier's conjecture. References [KOP09] Tadeusz Krasiński, Grzegorz Oleksik and Arkadiusz Płoski. The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity. Proc. Amer. Math. Soc., 137(10):3387-3397, 2009. [TYZ10] Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo. Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity. Proc. Amer. Math. Soc., 138(11):3975-3984, 2010.