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On splitting up singularities of fundamental solutions to elliptic equations in ℂ2

T. Savina (2007)

Open Mathematics

It is known that the fundamental solution to an elliptic differential equation with analytic coefficients exists, is determined up to the kernel of the differential operator, and has singularities on characteristics of the equation in ℂ2. In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.

On the approximate roots of polynomials

Janusz Gwoździewicz, Arkadiusz Płoski (1995)

Annales Polonici Mathematici

We give a simplified approach to the Abhyankar-Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.

On the diffeomorphic type of the complement to a line arrangement in a projective plane

Fedor Bogomolov, Viktor Kulikov (2012)

Open Mathematics

We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011,...

On the Euler characteristic of the links of a set determined by smooth definable functions

Krzysztof Jan Nowak (2008)

Annales Polonici Mathematici

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ( C ) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

On the Euler characteristic of the real Milnor fibres of an analytic function

Piotr Dudziński (2003)

Annales Polonici Mathematici

The paper is concerned with the relations between real and complex topological invariants of germs of real-analytic functions. We give a formula for the Euler characteristic of the real Milnor fibres of a real-analytic germ in terms of the Milnor numbers of appropriate functions.

On the fiber of the compound of a real analytic function by a projection

Alain Jacquemard (1999)

Bollettino dell'Unione Matematica Italiana

Sia f : R m R k con m k 1 una funzione analitica. Se il luogo critico di f è compatto, esiste una fibrazione C localmente triviale associata ai livelli f . Supponiamo k 2 e sia π k la proiezione x 1 , , x k - 1 , x k x 1 , , x k - 1 , x k . Sotto una condizione sul luogo critico di f ~ = π k f esiste anche una fibrazione C localmente triviale associata ai livelli di f ~ . Siano F e F ~ le fibre rispettitive, e I l'intervallo unità reale. Dimostriamo qui che F ~ è omeomorfa al prodotto F × I . Nel caso di polinomi studiamo criteri effettivi. Diamo inoltre un'applicazione del risultato...

On the geometry of polynomial mappings at infinity

Anna Valette, Guillaume Valette (2014)

Annales de l’institut Fourier

We associate to a given polynomial map from 2 to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Currently displaying 441 – 460 of 785