TY - JOUR
TI - On the gradient of quasi-homogeneous polynomials
AU - Haraux, Alain
AU - Pham, Tien Son
TI - On the gradient of quasi-homogeneous polynomials
AB - 
	Let K be the real or the complex field, and let f : Kn → K be a quasi-homogeneous polynomial with weight w := (w1;w2;...,wn) and degree d. Assume that rf(0) = 0. Łojasiewicz well known gradient inequality states that there exists an open neighbourhood U of the origin in Kn and two positive constants c and p < 1 such that for any x → U we have rf(x) > cf(x)p: We prove that if the set K - (f) of points where the Fedoryuk condition fails to hold is nite, then the gradient inequality holds true with p = 1-minj wj/d. It is also shown that if n = 2; then K-(f) is either empty or reduced to {0}.
VL - 2011
IS - Tom 49
PY - 2012
SN - 0083-4386
C1 - 2084-3828
SP - 45
EP - 57
DO - 10.4467/20843828AM.12.003.0455
UR - https://ejournals.eu/czasopismo/universitatis-iagellonicae-acta-mathematica/artykul/on-the-gradient-of-quasi-homogeneous-polynomials
KW - Quasi-homogeneous
KW - Lojasiewicz's gradient inequality
KW - Lojasiewicz exponent
KW - Fedoryuk's condition