Necessary conditions for irreducibility of algebroid plane curves

Let K be an algebraically closed field of characteristic 0 and let ƒ ϵ K[[X]] [Y] be monic. Using the properties of approximate roots given in [J. Algebra 343 (2011), pp. 143-159] we propose some necessary conditions for irreducibility of ƒ in K[[X]] [Y]. The result is expressed only in terms of intersection multiplicities of ƒ with its approximate roots

Degenerate singularities and their Milnor numbers

We give an example of a curious behaviour of the Milnor number with respect to evolving degeneracy of an isolated singularity in C2

A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularities

We prove that in order to find the value of the Łojasiewicz exponent ł(f) of a Kouchnirenko non-degenerate holomorphic function f : (Cn; 0) → (C; 0) with an isolated singular point at the origin, it is enough to find this value for any other (possibly simpler) function g : (Cn; 0) → (C; 0), provided this function is also Kouchnirenko non-degenerate and has the same Newton diagram as f does. We also state a more general problem, and then reduce it to a Teissier-like result on (c)-cosecant deformations, for formal power series with coefficients in an algebraically closed field K

Zariski multiplicity conjecture in families of non-degenerate singularities

Podajemy nowy, elementarny dowód hipotezy o krotności Zariskiego w μ-constant rodzinach niezdegenerowanych osobliwości

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

Milnor Numbers of Deformations of Semi-Quasi-Homogeneous Plane Curve Singularities

The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularity f. Assuming that f is irreducible, one can write f=∑qα+pβ ≥ pqcαβ xαyβ where cp0c0q≠0, 2≤p<q and p, q are coprime. We show that as Milnor numbers of deformations of f one can attain all numbers from μ(f) to μ(f)−r(p−r), where q≡r(mod p). Moreover, we provide an algorithm which produces the desired deformations