Invertible polynomial mappings via Newton non-degeneracy

Abstract

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers

“…Since, by Hadamard's above-mentioned theorem, the non-vanishing property of det JF actually is necessary for the global diffeomorphism property of F , it is thus an interesting question which additional conditions imposed on F , general enough, can assure that F is a global diffeomorphism of R n onto itself. Answering this question, which is also posed by Bivià-Ausina in [7] and which is of significant importance in [10] as well, is the main motivation for the present article. From Hadamard's theorem it is clear that such additional conditions must be related to the properness of F .…”

Toward Effective Detection of the Bifurcation Locus of Real Polynomial Maps

“…Since ν is a semi-algebraic mapping (see e.g [16,Proposition 2.4]), the Curve Selection Lemma and (11) imply that there there exists an analytic path (see also the proofs of [6,Theorem 3.2] and [3,Proposition 2.4] for this argument):…”

On stability and the Łojasiewicz exponent at infinity of coercive polynomials

“…Also, since coercivity of f is equivalent to the boundedness of its lower level sets {x ∈ R n | f (x) ≤ α} for all α ∈ R, understanding coercivity can be useful to decide whether a basic semialgebraic set is bounded. Furthermore, properness of polynomial maps F : R n → R n can be characterized by coercivity of the polynomial F 2 2 , which, can be used to decide whether F is globally invertible and it directly refers to real versions of the Jacobian conjecture (see, e. g. [3,6,8]).…”