25 research outputs found
The Łojasiewicz exponent over a field of arbitrary characteristic
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
Näkymätön tekijä
Let be an algebraically closed field of any characteristic. We apply the
Hamburger-Noether process of successive quadratic transformations to show the
equivalence of two definitions of the {\L}ojasiewicz exponent
of an ideal