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

Non-characteristic approximate roots of polynomials

AbstractWe generalize Abhyankar–Mohʼs theory of approximate roots of polynomials to the case of approximate roots of non-characteristic degrees of an irreducible element of K((X))[Y]

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

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

Three Decision-making Mechanisms to facilitate Negotiation of Service Level Agreements for Web Service Compositions

The negotiation of Service Level Agreements for composite web services is a very complex process. It involves the coordination of the negotiation process so that the end-to-end QoS requirements of the user request are satisfied while ensuring that the atomic QoS requirements are also simultaneously satisfied. This paper summarizes three decision-making mechanisms which support the process of Service Level Agreement negotiation for composite web services. The mechanisms include: the decomposition of the overall user preferences into the preferences of individual negotiation agents representing each atomic services within the composition; the selection of the prospective negotiation partners for the actual interaction from a list of potential service providers and finally the negotiation of Service Level Agreement with the selected provider agents while ensuring that the end-to-end QoS is satisfied

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