27 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
Processing of functional fine scale ceramic structures by ink-jet printing
International audienceThis review illustrates the potentiality of ink-jet printing for the fabrication of functional fine scale ceramic structures corresponding to two different kinds of micro-pillar arrays i.e. (i) PZT skeletons, etc..
Characterizing normal crossing hypersurfaces
The objective of this article is to give an effective algebraic
characterization of normal crossing hypersurfaces in complex manifolds. It is
shown that a hypersurface has normal crossings if and only if it is a free
divisor, has a radical Jacobian ideal and a smooth normalization. Using K.
Saito's theory of free divisors, also a characterization in terms of
logarithmic differential forms and vector fields is found and and finally
another one in terms of the logarithmic residue using recent results of M.
Granger and M. Schulze.Comment: v2: typos fixed, final version to appear in Math. Ann.; 24 pages, 2
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