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Displaying similar documents to “On the Łojasiewicz exponent near the fibre of polynomial mappings”

On the complexification of real-analytic polynomial mappings of ℝ²

Ewa Ligocka (2006)

Annales Polonici Mathematici

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We give a simple algebraic condition on the leading homogeneous term of a polynomial mapping from ℝ² into ℝ² which is equivalent to the fact that the complexification of this mapping can be extended to a polynomial endomorphism of ℂℙ². We also prove that this extension acts on ℂℙ²∖ℂ² as a quotient of finite Blaschke products.

The Łojasiewicz gradient inequality in a neighbourhood of the fibre

Janusz Gwoździewicz, Stanisław Spodzieja (2005)

Annales Polonici Mathematici

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Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.

Mappings of degree 5, part I

M. Maciejewski, A. Prószyński (2009)

Colloquium Mathematicae

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The class of linear (resp. quadratic) mappings over a commutative ring is determined by a set of equation-type relations. For the class of homogeneous polynomial mappings of degree m ≥ 3 it is so over a field, and over a ring there exists a smallest equationally definable class of mappings containing the preceding one. It is proved that generating relations determining that class can be chosen to be strong relations (that is, of the same form over all commutative rings) if{f} m ≤ 5....

Confluent mappings of fans

J. J. Charatonik, W. J. Charatonik, S. Miklos

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CONTENTS1. Introduction.......................................................52. Preliminaries ....................................................83. General properties .........................................114. Mappings onto fans........................................145. Mappings onto an arc.....................................206. A characterization of the top...........................277. Open mappings and their lightness................288. Inverse limits...................................................399....

On the extendability of quadratic polynomial mappings of the plane

Ewa Ligocka (2009)

Annales Polonici Mathematici

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We shall prove, using the result from our previous paper [Ann. Polon. Math. 88 (2006)], that for a quadratic polynomial mapping Q of ℝ² only the geometric shape of the critical set of Q determines whether the complexification of Q can be extended to an endomorphism of ℂℙ². At the end of the paper we describe some interesting classes of quadratic polynomial mappings of ℝ² and give some examples.

Combinatorial Nullstellensatz approach to polynomial expansion

Fedor Petrov (2014)

Acta Arithmetica

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Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of |f(A,B)| for finite subsets A, B of a field, and a polynomial f(x,y) of the form f(x,y) = g(x) + yh(x), where the degree of g is greater than that of h.