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

We give some estimations of the Łojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the Łojasiewicz exponent of such singularities in some special cases. The results are stronger than Fukui inequality [8]. It is also a multidimensional generalization of the Lenarcik theorem [13].

The Łojasiewicz exponent of nondegenerate surface singularities

The Łojasiewicz inequality and Łojasiewicz exponent reveal a fundamental relation between a potential function and its gradient. In this paper, we explore the Łojasiewicz exponent of Kuramoto model and prove that the exponent is exactly 12 for equilibria located inside a quarter of circle. This implies that the convergence towards such a phase-locked state must be exponentially fast. In contrast, we give an example to see the exponent can be less than 12 for other equilibriums. More precisely, we prove that the exponent for the bi-cluster equilibrium, which is located on the boundary of a quarter of circle, is 13. This gives an insight for the occurrence of exponential and algebraic convergence of Kuramoto model. We also present a general theorem for exponential convergence of second-order gradient-like system, by which a criterion for the Kuramoto model with inertia is established.

On the Łojasiewicz exponent of Kuramoto model

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.

/pdf/on-combinatorial-criteria-for-non-degenerate-singularities-38wzl3mnw5.pdf

On combinatorial criteria for non-degenerate singularities

Mei-Chi Shaw) We give an explicit formula for the Lojasiewicz exponent of an isolated weighted homogeneous surface singularity in terms of its weights. From the formula we get that the Lojasiewicz exponent is a topological invariant of these singularities.

/pdf/the-lojasiewicz-exponent-of-an-isolated-weighted-homogeneous-3flf23p3qr.pdf

The Lojasiewicz exponent of an isolated weighted homogeneous surface singularity

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

In this article, we show that combining random generators by group operation improves the statistical properties of the composite. It gives an effective way of finding random generators more and more close to the uniform. Moreover, we obtain an effective estimation of the speed of convergence to the uniform generator.

/pdf/combining-random-generators-by-group-operation-3k2u2pjstc.pdf

Combining random generators by group operation

Let f be a real polynomial, non-negative at infinity with non-compact zero-set. Suppose that f is non-degenerate in the Kushnirenko sense at infinity. In this paper we give a formula for the Łojasiewicz exponent at infinity of f and a formula for the exponent of growth of f in terms of its Newton polyhedron.

/pdf/the-lojasiewicz-exponent-at-infinity-of-non-negative-and-non-477q69o8ym.pdf

Grzegorz Oleksik

Papers

The Lojasiewicz exponent of an isolated weighted homogeneous surface singularity

On combinatorial criteria for non-degenerate singularities

A conjecture on the Łojasiewicz exponent

Combining random generators by group operation

The Łojasiewicz Exponent at Infinity of Non-negative and Non-degenerate Polynomials