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On the Łojasiewicz Exponents of Quasi-Homogeneous Functions

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On the Łojasiewicz Exponents of Quasi-Homogeneous Functions Journal of Singularities received: 29 June 2013 Volume 11 (2015), 52-66 in revised form: 16 April 2015 DOI: 10.5427/jsing.2015.11c ON THELOJASIEWICZ EXPONENTS OF QUASI-HOMOGENEOUS FUNCTIONS ALAIN HARAUX AND TIEN SON PHA. M Abstract. Quasi-homogeneous functions, and especially polynomials, enjoy some specific properties around the origin which allow to estimate the so-calledLojasiewicz exponents in a way quite similar to homogeneous functions. In particular we generalize a previous result for polynomials of two variables concerning the optimalLojasiewicz gradient inequality at the origin. 1. Introduction In his pioneering papers [13], [14],Lojasiewicz established that any analytic function f of n real variables satisfies an inequality of the form (1.1) krf(x)k ≥ cjf(x) − f(a)jβ for kx − ak small enough with c > 0; β 2 (0; 1). This inequality, known as theLojasiewicz gradient inequality, is useful to establish trend to equilibrium of the general solutions of gradient systems (1.2) u0 + rf(u) = 0 and can also be used to evaluate the rate of convergence. It is therefore of interest to know as precisely as possible the connection between f and its gradient and in particular to determine the best(smallest) possible value of β in (1.1) when a is a critical point of f. This value is called theLojasiewicz exponent at a: In [7] for instance, it was shown that if f is a homogeneous 1 polynomial with degree d ≥ 2, theLojasiewicz exponent at the origin is exactly 1 − d when n = 2. This property is no longer true if n > 2. On the other hand, Gwo´zdziewicz[6] (see also [16]) considered the case of a real analytic function at an isolated zero and also found, in this case, an interesting relationship between variousLojasiewicz exponents, relative to differentLojasiewicz inequalities. In addition the case of general polynomials has been thoroughly investigated by D'Acunto and Kurdyka in [2] (see also [11, 12, 17]). Our paper is concerned to the extension of the result from [7] and several estimates of Lojasiewicz exponents at the origin when f is a quasi-homogeneous map (see, for example, [1]). It is divided in 5 sections. In Section2, we state and prove some preliminary results, mainly concerning the local behavior of quasi-homogeneous maps near the origin. Section3 contains more information in the specific case where the origin is an isolated zero of f: Section 4 deals with theLojasiewicz gradient exponent of quasi-homogeneous polynomials, in particular we generalize the main result of [7] (see also [4,5,9] for related results). Section5 contains 2010 Mathematics Subject Classification. 14P10, 14P15, 32S05, 34A26. Key words and phrases. quasi-homogeneous, gradient inequality,Lojasiewicz exponents. The second author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant number 101.04-2013.07. ON THELOJASIEWICZ EXPONENTS OF QUASI-HOMOGENEOUS FUNCTIONS 53 more precise estimates for quasi-homogeneous polynomials of two variables. These results are illustrated by typical examples and completed by a few remarks. 2. Definitions and preliminary results We first recall the concept of a quasi-homogeneous map. Let K = C or R: We say that f : Kn ! Kk is a (positively) quasi-homogeneous map with weight n w := (w1; w2; : : : ; wn) 2 (R+ − f0g) k and quasi-degree d := (d1; d2; : : : ; dk) 2 (R+ − f0g) if w1 w2 wn di (2.1) fi(t x1; t x2; : : : ; t xn) = t fi(x1; x2; : : : ; xn) for each i = 1; 2; : : : ; k; and all t > 0: Note that if wj = 1 for j = 1; 2; : : : ; n then the above definition means that the components fi are homogeneous functions of degree di: When k = 1, a scalar function f = f1 satisfying (2.1) is called a (positively) quasi-homogeneous function with weight w = (w1; w2; : : : ; wn) and quasi-degree d = d1: In the sequel we shall drop for α α1 α2 αn simplicity the word \positively". Note that any monomial x := x1 x2 : : : xn is a quasi- n homogeneous function with arbitrary weight w = (w1; w2; : : : ; wn) 2 (N − f0g) and quasi- degree hw; αi := w1α1 + w2α2 + ··· + wnαn: Moreover, we have Proposition 2.1. Let f : Kn ! K be a polynomial function. Then f is quasi-homogeneous with weight w 2 (N − f0g)n and quasi-degree m 2 N − f0g if and only if all its constitutive monomials are quasi-homogeneous functions with weight w and quasi-degree m: Proof. Suppose that f is a quasi-homogeneous polynomial with weight w and quasi-degree m: We have the following finite expansion X α f(x) := aαx : α Then X hw,αi α m X α aαt x = t aαx : α α This gives X hw,αi−m α aα[t − 1]x = 0 α for all x 2 Kn and for all t > 0: Since the field K is of characteristic 0, we get hw; αi − m = 0 for all α provided aα 6= 0: In other words, all constitutive monomials of f are quasi-homogeneous functions with weight w and quasi-degree m: The converse is clear. n For a fixed weight w := (w1; w2; : : : ; wn) 2 (R+ − f0g) we set 1 w kxkw := max jxjj j j=1;2;:::;n n 1 for x := (x1; x2; : : : ; xn) 2 K : In the special case w = w0 = (1; :::1) , we recover the usual l norm on Kn and we set kxk := kxkw = max jxjj: 0 j=1;2;:::;n 54 ALAIN HARAUX AND TIEN SON PHA. M Remark 2.2. (i) It is worth noting that, except in the special case w = w0 = (1; :::1), k · kw is not a norm since it is not homogeneous of degree 1. (ii) It is easy to see that k · kw is a quasi-homogeneous function with weight w and with quasi-degree 1: The following basic properties will be used throughout the text. Proposition 2.3. Let w∗ := min wj; j=1;2;:::;n ∗ w := max wj: j=1;2;:::;n Then the following hold (i) For all kxk ≥ 1 we have 1 1 ∗ kxk w∗ ≥ kxkw ≥ kxk w : In particular, kxk ! 1 if and only if kxkw ! 1: (ii) For all kxk ≤ 1 we have 1 1 ∗ kxk w∗ ≤ kxkw ≤ kxk w ; In particular, kxk ! 0 if and only if kxkw ! 0: Proof. The proof of the proposition is straightforward from the definitions. n In the sequel for t > 0, for any w := (w1; w2; : : : ; wn) 2 (R+ − f0g) and n x := (x1; x2; : : : ; xn) 2 K we denote w1 w2 wn t • x := (t x1; t x2; : : : ; t xn); k and for d := (d1; d2; : : : ; dk) 2 (R+ − f0g) we set d∗ := min di; i=1;2;:::;k ∗ d := max di: i=1;2;:::;k n k n Let f : K ! K be quasi-homogeneous with weight w := (w1; w2; : : : ; wn) 2 (R+ − f0g) k and quasi-degree d := (d1; d2; : : : ; dk) 2 (R+ − f0g) . If fi ≡ 0 for some i 2 f1; 2; : : : ; kg; then di can be replaced by any positive number. In the sequel we shall assume (2.2) 8i 2 f1; 2; : : : ; kg; fi 6≡ 0: It is easy to check that in this case di is uniquely defined by (2.1) for all i 2 f1; 2; : : : ; kg: Then ∗ d∗ and d are well defined. The next two results summarize some important consequences of the quasi homogeneity prop- erty. n k Proposition 2.4. Let f := (f1; f2; : : : ; fk): K ! K be a quasi-homogeneous map with weight w := (w1; w2; : : : ; wn) and quasi-degree d := (d1; d2; : : : ; dk) satisfying (2.2). Then the following properties are equivalent. (i) The origin is an isolated zero of f. (ii) f −1(0) = f0g: ON THELOJASIEWICZ EXPONENTS OF QUASI-HOMOGENEOUS FUNCTIONS 55 Proof. It is clear that (ii) implies (i). Conversely if (ii) is not satisfied, let a 6= 0 be such that f(a) = 0: This implies that f1(a) = f2(a) = ··· = fk(a) = 0: Hence di fi(t • a) = t fi(a) = 0 for i = 1; 2; : : : ; k and all t > 0: Note that kt • ak ! 0 as t ! 0: Thus, the origin is not an isolated zero of f; which is a contradiction. Proposition 2.5. Let f : Rn ! R be a continuous quasi-homogeneous function with weight w and quasi-degree m: Suppose that n ≥ 2: Then the following conditions are equivalent (i) f −1(0) = f0g: (ii) f has a strict global extremum at the origin. (iii) For each ≥ 0; fx 2 Rn j jf(x)j = g is a non-empty compact set. (iv) minkxk=1 jf(x)j > 0: Proof. (i) ) (ii) If f −1(0) = f0g; then by connectedness f has a constant sign (for instance f > 0) on the unit Euclidian sphere Sn−1 := fx 2 Rn j kxk = 1g of dimension (n − 1) if n > 1: But for any x 6= 0; there is clearly t > 0 such that y := t • x is in Sn−1: Indeed the euclidian norm of t • x is 0 for t = 0; tends to infinity with t and is a continuous function of t; hence it must take the value 1 for some finite positive t: Then f(x) = f(t−1 • y) = t−mf(y) > 0; which proves (ii). (ii) ) (iv) Suppose, by contradiction, that minkxk=1 jf(x)j = 0: Then there exists a point a 2 Rn such that kak = 1 and f(a) = 0: This implies that f(t • a) = 0 for all t > 0; which contradicts (ii). (iv) ) (iii) By contradiction, assume that the set fx 2 Rn j jf(x)j = g is not compact for some ≥ 0: This means that there exists a sequence xp 2 Rn; p 2 N; such that kxpk ! 1 as p 1 p m p ! 1 and jf(x )j = .
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