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Łojasiewicz Exponent of Overdetermined Mappings

Stanisław Spodzieja, Anna Szlachcińska (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

A mapping F : ℝ ⁿ → ℝ m is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping F : ℝ ⁿ → ℝ m can be reduced to the case m = n.

Łojasiewicz exponent of the gradient near the fiber

Ha Huy Vui, Nguyen Hong Duc (2009)

Annales Polonici Mathematici

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber f - 1 ( t ₀ ) such that r is the Łojasiewicz exponent of grad(f) near the fiber f - 1 ( t ₀ ) . We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

Janusz Gwoździewicz, Arkadiusz Płoski (2005)

Colloquium Mathematicae

For every polynomial F in two complex variables we define the Łojasiewicz exponents ℒ p , t ( F ) measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents ℒ p , t ( F ) in terms of the local invariants of singularities of the pencil of projective curves associated with F.

[unknown]

Raimundo Araújo dos Santos, Maria A.B. Hohlenwerger, Osamu Saeki, Taciana O. Souza (0)

Annales de l’institut Fourier

μ -constant monodromy groups and marked singularities

Claus Hertling (2011)

Annales de l’institut Fourier

μ -constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo ± id . Second, marked singularities are defined and global moduli spaces for right equivalence...

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