scholarly journals Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity

2010 ◽  
Vol 138 (11) ◽  
pp. 3975-3975 ◽  
Author(s):  
Shengli Tan ◽  
Stephen S.-T. Yau ◽  
Huaiqing Zuo
Keyword(s):  
Homogeneous Polynomial ◽  
Isolated Singularity ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality ◽  
Weighted Homogeneous Polynomial

scholarly journals THE ŁOJASIEWICZ EXPONENT FOR WEIGHTED HOMOGENEOUS POLYNOMIAL WITH ISOLATED SINGULARITY

2016 ◽  
Vol 59 (2) ◽  
pp. 493-502
Author(s):  
OULD M. ABDERRAHMANE
Keyword(s):  
Explicit Formula ◽  
Homogeneous Polynomial ◽  
Isolated Singularity ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent ◽  
Weighted Homogeneous Polynomial

AbstractThe purpose of this paper is to give an explicit formula of the Łojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

Łojasiewicz inequality for a pair of semialgebraic functions

2021 ◽  
Vol 166 ◽  
pp. 102927
Author(s):  
Beata Osińska-Ulrych ◽  
Grzegorz Skalski ◽  
Anna Szlachcińska
Keyword(s):  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

ŁOJASIEWICZ INEQUALITY FOR POLYNOMIAL FUNCTIONS ON NON-COMPACT DOMAINS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250033 ◽  
Author(s):  
DINH SI TIEP ◽  
HA HUY VUI ◽  
NGUYEN THI THAO
Keyword(s):  
Polynomial Functions ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

In this paper we give some versions of the Łojasiewicz inequality on non-compact domains for polynomial functions. We also point out some relations between the existence Łojasiewicz inequality and the phenomenon of singularities at infinity.

scholarly journals Topological invariants of weighted homogeneous polynomials

1991 ◽  
Vol 33 (3) ◽  
pp. 241-245 ◽  
Author(s):  
Zbigniew Szafraniec
Keyword(s):  
Euler Characteristic ◽  
Homogeneous Polynomial ◽  
Topological Invariants ◽  
Homogeneous Polynomials ◽  
Weighted Homogeneous Polynomial

Let ℝn → ℝ be a weighted homogeneous polynomial such that df(0) = 0, L = {x ∈ Sn−1|f(x) = 0}, and let χ(L) be the Euler characteristic of L. The problem is how to calculate χ(L) in terms of f.

scholarly journals Topological invariants of germs of real analytic functions

1997 ◽  
Vol 39 (1) ◽  
pp. 85-89 ◽  
Author(s):  
Piotr Dudziński
Keyword(s):  
Real Analytic Function ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Isolated Singularity ◽  
Simple Consequence ◽  
Milnor Fibre ◽  
Real Analytic Functions ◽  
Nonisolated Singularity ◽  
Real Analytic ◽  
Weighted Homogeneous Polynomial

Let f: (ℝn, 0)→ (ℝ,0) be a germ of a real analytic function. Let L and F(f) denote the link of f and the Milnor fibre of fc respectively, i. e., L = {x ∈ Sn−1 | f(x) = 0}, , where 0 ≤ ξ ≪ r ≪ 1, . In [2] Szafraniec introduced the notion of an -germ as a generalization of a germ defined by a weighted homogeneous polynomial satisfying some condition concerning the relation between its degree and weights (definition 1). He also proved that if f is an -germ (presumably with nonisolated singularity) then the number χ(F(f)/d mod 2 is a topological invariant of f, where χ(F(f)) is the Euler characterististic of F(f), and gave the formula for χ(L)/2 mod 2 (it is a well-known fact that F(L) is an even number). As a simple consequence he got the fact that χ(F(f)mod 2 is a topological invariant for any f, which is a generalization of Wall's result [3] (he considered only germs with an isolated singularity).

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