Description

• Title:
  The Łojasiewicz exponent over a field of arbitrary characteristic
• Creator:
  Brzostowski, Szymon;
  Rodak, Tomasz
• Contributor:
  University of Łódź, Faculty of Mathematics and Computer Science;
  brzosts@math.uni.lodz.pl;
  rodakt@math.uni.lodz.pl
• Description:
  Let K be an algebraically closed field and let K((XQ)) denote the field of generalized series with coefficients in K. We propose definitions of the local Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 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((XQ))[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.
• Object availability:
  free access
• Rights:
  Uznanie autorstwa 3.0 Polska;
  http://creativecommons.org/licenses/by/3.0/pl
• Date:
  2015.07.30;
  2015.01.13
• Type:
  Article
• Language:
  English
• Format:
  application/pdf
• Publisher:
  Springer Verlag
• Relation:
  Revista Matemática Complutense;28
• Subject:
  Generalized power series;
  Łojasiewicz exponent;
  Parametrization;
  Newton polygon method
• Identifier:
  1988-2807;
  http://hdl.handle.net/11089/11274 ;
  10.1007/s13163-014-0165-3
• Data provider:
  Repozytorium Uniwersytetu Łódzkiego
• Can I use it?:
  check license
• Type:
  image