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Justyna Walewska

Publications and source records attributed to Justyna Walewska.

7 recordsLinked to original sources
arXiv2018

Jumps of Milnor numbers of Brieskorn-Pham singularities in non-degenerate families

The jump of the Milnor number of an isolated singularity $f_{0}$ is the minimal non-zero difference between the Milnor numbers of $f_{0}$ and one of its deformation $(f_{s}).$ In the case $f_{s}$ are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of $f_{0}$ in the case $f_{0}$ is a convenient singularity with only one $(n-1)$-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn-Pham singularities.

math.AG↗
arXiv2017

Arnold's problem on monotonicity of the Newton number for surface singularities

According to the Kouchnirenko theorem, for a generic (precisely non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $μ(f)$ is equal to the Newton number $ν(Γ_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $Γ_+ (f)$. We give a simple condition characterising, in terms of $Γ_+ (f)$ and $Γ_+ (g)$, the equality $ν(Γ_{+}(f)) = ν(Γ_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $Γ_+ (f) \subset Γ_+ (g)$. This is a complete solution to an Arnold's problem (1982-16) in this case.

math.AG↗
arXiv2016

Milnor numbers in deformations of homogeneous singularities

Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.

math.AG↗
arXiv2016

Non-degenerate jump of Milnor numbers of surface singularities

The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)$. We give a formula for the jump in some class of surface singularities in the case deformations are non-degenerate.

math.AG↗
arXiv2014

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 singularities. Main result states that if $f$ is irreducible and nondegenerate, by deforming $f$ one can attain all Milnor numbers ranging from $μ(f)$ to $μ(f)-r(p-r)$, where $r$ and $p$ are easily computed from the Newton diagram of $f$.

math.AG↗