On the Łojasiewicz exponent of the gradient of a holomorphic function

Abstract

Abstract. The Lojasiewicz exponent of the gradient of a convergent power series h(X, Y ) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality |grad h(x, y)| ≥ c|(x, y)| λ holds near 0 ∈ C 2 for a certain c > 0. In the paper, we give an estimate of the Lojasiewicz exponent of grad h using information from the Newton diagram of h.We obtain the exact value of the exponent for non-degenerate series.1. Introduction. The main goal of this paper is to compute the Lojasiewicz exponent of the plane curve singularity using the Newton Polygon. Our theorem is the counterpart of the Kouchnirenko theorem [Kou] on the Milnor number in the case of two variables. We were inspired by the articles of Lichtin [Li] and Fukui [Fu] who proved some estimates for the Lojasiewicz exponent. However, our considerations are based on other ideas and enable us to calculate the Lojasiewicz exponent for non-degenerate power series. The organization of the paper is as follows. In Section 2 we present the main theorem with illustrative examples. Then we collect in Section 3 basic notions connected with the Newton diagrams. Our main reference is [BK]. A theorem concerning the Lojasiewicz exponent of a holomorphic mapping, determined by the pair of convergent power series, is given in Section 4. The theorem is helpful in a proof of the main result given in Section 6. The proof is preceded by the theorems which describe connections between the Newton diagram of the series h and the diagrams of its derivatives

Jacobian Ideals and the Newton Non-Degeneracy Condition

“…This is defined as the infimum of those real numbers α > 0 such that there There are many works dealing with the computation of the number α 0 (f ) (see [1,12,17,22]). Let J(f ) denote the Jacobian ideal of a given f ∈ O n , that is, the ideal of O n generated by the partial derivatives ∂f /∂x 1 , .…”

Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals

“…We remark that if I 1 , I 2 are two monomial ideals of O 2 such that σ (I 1 , I 2 ) < ∞ and if g = (g 1 , g 2 ) : (C 2 , 0) → (C 2 , 0) is a finite analytic map such that + (g i ) = + (I i ), for i = 1, 2, then g is non degenerate with respect to I 1 , I 2 if and only if the map g satisfies the condition given by Lenarcik in [20,Definition 4.1]. Therefore [20,Theorem 4.2] shows an effective computation of L 0 (g), for all g ∈ R 0 (I 1 , I 2 ) in terms of certain combinatorial aspects of + (I 1 ) and + (I 2 ) that are easily computable (see also [9,Theorem 4.3]).…”

“…Therefore [20,Theorem 4.2] shows an effective computation of L 0 (g), for all g ∈ R 0 (I 1 , I 2 ) in terms of certain combinatorial aspects of + (I 1 ) and + (I 2 ) that are easily computable (see also [9,Theorem 4.3]). The techniques applied in the proof of the said result of Lenarcik for maps of two complex variables are based on the Newton-Puiseux theorem.…”

“…The computation or estimation from above of L 0 ( f ) is not straightforward at all. We refer to [6,14] or [20] for results about this problem that consider the information supplied by the Newton polyhedron of f .…”

The Łojasiewicz Exponent for Weighted Homogeneous Polynomial With Isolated Singularity

“…An interesting mathematical problem is to give formulas for L(f ) in terms of another invariants of f or an algorithm to compute it. In the two-dimensional case there are many explicit formulas for L(f ) in various terms (see [4], [5], [9], [13]). Estimations of the Lojasiewicz exponent in the general case can be found in [1], [6], [14], [19].…”