Displaying similar documents to “On the Łojasiewicz exponent at infinity of real polynomials”

Local polynomials are polynomials

C. Fong, G. Lumer, E. Nordgren, H. Radjavi, P. Rosenthal (1995)

Studia Mathematica

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We prove that a function f is a polynomial if G◦f is a polynomial for every bounded linear functional G. We also show that an operator-valued function is a polynomial if it is locally a polynomial.

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.

The degree at infinity of the gradient of a polynomial in two real variables

Maciej Sękalski (2005)

Annales Polonici Mathematici

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Let f:ℝ² → ℝ be a polynomial mapping with a finite number of critical points. We express the degree at infinity of the gradient ∇f in terms of the real branches at infinity of the level curves {f(x,y) = λ} for some λ ∈ ℝ. The formula obtained is a counterpart at infinity of the local formula due to Arnold.

Plane Jacobian conjecture for simple polynomials

Nguyen Van Chau (2008)

Annales Polonici Mathematici

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A non-zero constant Jacobian polynomial map F=(P,Q):ℂ² → ℂ² has a polynomial inverse if the component P is a simple polynomial, i.e. its regular extension to a morphism p:X → ℙ¹ in a compactification X of ℂ² has the following property: the restriction of p to each irreducible component C of the compactification divisor D = X-ℂ² is of degree 0 or 1.