Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro.
Sottile, Frank (2000)
Experimental Mathematics
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Sottile, Frank (2000)
Experimental Mathematics
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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.
Bjorn Poonen (2010)
Acta Arithmetica
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L. Andrew Campbell (2014)
Annales Polonici Mathematici
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Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples...
Ken McKenna, Lou Van den Dries (1990)
Manuscripta mathematica
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Ludwik Drużkowski (2005)
Control and Cybernetics
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Frederico Xavier (1993)
Mathematische Annalen
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Adam H. Piwocki (2007)
Colloquium Mathematicae
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We study the determinant of pairs of rotants of Anstee, Przytycki and Rolfsen. We consider various notions of rotant orientations.
Luís R. A. Finotti (2009)
Acta Arithmetica
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Umberto Zannier (2007)
Acta Arithmetica
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Tomasz Rodak (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
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We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.
R. Ger (1971)
Annales Polonici Mathematici
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Jason Lucier (2006)
Acta Arithmetica
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Norbert Hegyvári, François Hennecart (2009)
Acta Arithmetica
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J. Siciak (1971)
Annales Polonici Mathematici
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Janusz Gwoździewicz (2001)
Annales Polonici Mathematici
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We show that every local polynomial diffeomorphism (f,g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism.
Arno van den Essen (2001)
Annales Polonici Mathematici
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We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.
Mira Bhargava (1964)
Collectanea Mathematica
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Michiel Bondt (2014)
Open Mathematics
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This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.