The Łojasiewicz exponent over a field of arbitrary characteristic

Abstract

Let K be an algebraically closed field and let K((X Q )) denote the field of generalized series with coefficients in K. We propose definitions of the local Łojasiewicz exponent of [191][192][193][194][195][196][197] 1997), and prove some basic properties of such numbers. Namely, we show that in both cases the exponent is attained on a parametrization of a component of F (Theorems 6 and 7), thus being a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent of F ∈ (K((X Q )) [Y ]) m for which we give a description of all the generalized series that extract the pseudoexponent, in terms of their jets. In particular, we show that there exist only finitely many jets of generalized series giving the pseudoexponent of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton's Polygon Method. The results are illustrated with some explicit examples.

The Łojasiewicz exponent via the valuative Hamburger-Noether process

“…Moreover, they proved that, with the help of the notion of integral closure of an ideal, the number L(a) may be seen algebraically. This is what we generalize below (see Theorem 1) partly answering [3,Question 2]. D'Angelo [6] introduced L(a) independently, as an order of contact of a.…”

“…The case of ideals in k[[x, y]], where k is as above, is due to the authors [3]. De Felipe, García Barroso, Gwoździewicz and Płoski [7] gave a shorter proof of this result; moreover, they answered [3,Question 1], by showing that L(a) is always a Farey number, i. e. a rational number of the form N + b/a, where N , a, b are integers such that 0 < b < a < N .…”

On the algebraicity of generalized power series

“…• The following papers make only passing references to [26] and/or [27], with no reference to specific results, and are thus unaffected: [1], [2], [9], [10], [14], [15], [18], [19], [21], [22], [23], [24], [25], [30], [31], [35], [38], [39]. • The following papers only reference [26] (either explicitly or via [27]) in the case where K = F p , and are thus unaffected: [3], [28], [32], [35].…”