Isolated singularities defined by weighted homogeneous polynomials

Let $f: (\\mathbb{C}^n,0) \\rightarrow (\\mathbb{C},0)$ be a semiquasihomogeneous function. We give a formula for the local {\\L}ojasiewicz exponent $\\mathcal{L}_{0}(f)$ of $f$, in terms of weights of $f$. In particular, in the case of a quasihomogeneous isolated singularity $f$, we generalize a formula for $\\mathcal{L}_{0}(f)$ of Krasi\\'nski, Oleksik and P{\\l}oski ([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. Krasi\\'nski. \nAs a consequence of our result, we get that the {\\L}ojasiewicz exponent is invariant in topologically trivial families of singularities coming from a quasihomogeneous germ. This is an affirmative partial answer to Teissier's conjecture. \nReferences \n[KOP09] Tadeusz Krasi\\'nski, Grzegorz Oleksik and Arkadiusz P{\\l}oski. The {\\L}ojasiewicz exponent of an isolated weighted homogeneous surface singularity. Proc. Amer. Math. Soc., 137(10):3387-3397, 2009. \n[TYZ10] Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo. {\\L}ojasiewicz inequality for weighted homogeneous polynomial with isolated singularity. Proc. Amer. Math. Soc., 138(11):3975-3984, 2010.

The Łojasiewicz exponent of semiquasihomogeneous singularities

In this article we give a sufficient and necessary condition for a Kouchnirenko non-degenerate holomorphic function to have an isolated singularity at 0 in terms of its support. As a corollary we give some useful sufficient conditions for singularity to be isolated.

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On combinatorial criteria for non-degenerate singularities

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Łojasiewicz (1959, 1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). Here we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is C1 with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for (real or complex) analytic varieties, that the gradient inequality for an arbitrary analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Łojasiewicz inequalities when a function is CN and generalized Morse–Bott of order N≥3; we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is C2 and Morse–Bott on a Banach space.

https://msp.org/gt/2019/23-7/gt-v23-n7-p02-s.pdf

Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Let $f: (\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ be a semiquasihomogeneous function. We give a formula for the local {\L}ojasiewicz exponent $\mathcal{L}_{0}(f)$ of $f$, in terms of weights of $f$. In particular, in the case of a quasihomogeneous isolated singularity $f$, we generalize a formula for $\mathcal{L}_{0}(f)$ of Krasi\'nski, Oleksik and P{\l}oski ([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. Krasi\'nski. 
As a consequence of our result, we get that the {\L}ojasiewicz 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 Krasi\'nski, Grzegorz Oleksik and Arkadiusz P{\l}oski. The {\L}ojasiewicz 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. {\L}ojasiewicz inequality for weighted homogeneous polynomial with isolated singularity. Proc. Amer. Math. Soc., 138(11):3975-3984, 2010.

The {\L}ojasiewicz Exponent of Semiquasihomogeneous Singularities

In this paper, we present a conjecture connecting the Łojasiewicz exponent of an isolated nondegenerate singularity with some geometrical characteristics of the Newton diagram associated with this singularity. We prove the conjecture for a class of surface singularities.

A conjecture on the Łojasiewicz exponent

We prove the constancy of the Łojasiewicz exponent in non-degenerate $\mu$-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.

The {\L}ojasiewicz exponent in non-degenerate deformations of surface singularities

Let $f$ be an isolated singularity at the origin of $\mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {\L}ojasiewicz exponent $\mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for generic surface singularities $f$, an effective formula for $\mathcal{L}_0 (f)$ in terms of the Newton polyhedron of $f$. This is a realization of one of Arnold's postulates.

Szymon Brzostowski

Papers

On combinatorial criteria for non-degenerate singularities

A conjecture on the Łojasiewicz exponent

The {\L}ojasiewicz exponent in non-degenerate deformations of surface singularities

The {\L}ojasiewicz exponent of non-degenerate surface singularities