Fibers of polynomial mappings over ℝn

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Abstract

We find sufficient conditions on a polynomial mapping f = (p1,⋯,pn): ℝn → ℝn to be surjective. One such a condition is the existence of a non-trivial solution of the induced homogeneous system of the equations Σj=1n(pj)αjgij=0, i = 1,⋯,n. Here αjϵℤ+,gij=ℝ[X1,⋯,Xn] and det(gij) never vanishes on ℝn. A conclusion that follows is that if Πj=1n(degpj) is an odd integer, then surjectivity f(ℝ)n = ℝn follows if the homogeneous system p1=⋯=pn=0 (p is the highest homogeneous component of p) has only the trivial solution. We also investigate mappings f for which the determinant of their Jacobian matrix, det J(f) never vanishes on ℝn. These polynomial mappings are in the core of the Real Jacobian Conjecture. One conclusion is that for such a local polynomial diffeomorphism the system pj∂pjXi=0, i = 1,⋯,n must have non-trivial solutions, and for any j = 1,⋯,n. Also, such a local diffeomorphism is surjective if the induced homogeneous system of σj=1nαj(pj)αj-1∂pj∂Xi=0, i = 1,⋯n, has only the trivial (zero) solution. These last two theorems give a new point of view on S. Pinchuk's solution of the Real Jacobian Conjecture. Other obvious applications of our results are for the existence of solutions of the corresponding polynomial equations in n unknowns over the real field, ℝ.

Original languageEnglish
Title of host publicationMathematical Methods and Computational Techniques in Science and Engineering II
EditorsNikos Bardis
PublisherAmerican Institute of Physics Inc.
ISBN (Electronic)9780735416987
DOIs
StatePublished - 27 Jul 2018
Event2nd International Conference on Mathematical Methods and Computational Techniques in Science and Engineering - Cambridge, United Kingdom
Duration: 16 Feb 201818 Feb 2018

Publication series

NameAIP Conference Proceedings
Volume1982
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Conference

Conference2nd International Conference on Mathematical Methods and Computational Techniques in Science and Engineering
Country/TerritoryUnited Kingdom
CityCambridge
Period16/02/1818/02/18

ASJC Scopus subject areas

  • Physics and Astronomy (all)

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