On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers. (English) Zbl 1405.14130
If a polynomial map \(F: \mathbb{R}^n \to \mathbb{R}^n\) is a diffeomorphism, then the Jacobian \(JF\) of \(F\) satisfies \(\det JF \neq 0\) everywhere. However, there exist polynomial maps \(F\) for which \(\det JF \neq 0\) everywhere even though \(F\) is not a diffeomorphism [S. Pinchuk, Math. Z. 217, No. 1, 1–4 (1994; Zbl 0874.26008)]. Thus having a nonvanishing Jacobian determinant on its own does not yield a characterization of polynomial diffeomorphisms of \(\mathbb{R}^n\). The article treats the question which further assumptions on the polynomial map \(F\) are needed to deduce from \(\det(F)\neq 0\) that \(F\) is a diffeomorphism.
J. Hadamard [C. R. Acad. Sci., Paris 142, 74–77 (1906; JFM 37.0672.01)] characterized the diffeomorphism property for general \(C^1\)-maps \(F\) by the topological condition of properness: A \(C^1\)-map \(F: \mathbb{R}^n \to \mathbb{R}^n\) is a diffeomorphism if and only if \(\det JF \neq 0\) everywhere and \(F\) is proper, i.e. the preimages of compact sets under \(F\) are compact. However, for polynomial maps a characterization in terms of the polynomial coefficients is desirable and more applicable than Hadamard’s abstract characterization. The authors tackle this objective by giving sufficient criteria for the properness of \(F\) in terms of the coefficients of \(\|F\|_2^2\), making use of the Newton Polytope of \(\|F\|_2^2\) and the fact that \(\|F\|_2^2\) is a Sums-of-Squares polynomial. Key ingredients are the notions of gem regularity and circuit numbers.
In previous work [SIAM J. Optim. 25, No. 3, 1542–1570 (2015; Zbl 1322.90092)], the authors exhibited sufficient and necessary conditions for the properness of \(F\) that are being used in this article. Thus the present work focuses on the discussion of a parameterized family of illustrative examples. There, a small gap in between the necessary and sufficient criteria for properness manifests: The properness of \(F\) is invariant under linear transformations \(F\mapsto F\circ A\) (with \(A \in \text{GL}_n(\mathbb{R})\)), but the given sufficient criterion is not. The authors treat one example where direct application of their criterion fails and give an explicit linear map \(A\) that makes it applicable.
J. Hadamard [C. R. Acad. Sci., Paris 142, 74–77 (1906; JFM 37.0672.01)] characterized the diffeomorphism property for general \(C^1\)-maps \(F\) by the topological condition of properness: A \(C^1\)-map \(F: \mathbb{R}^n \to \mathbb{R}^n\) is a diffeomorphism if and only if \(\det JF \neq 0\) everywhere and \(F\) is proper, i.e. the preimages of compact sets under \(F\) are compact. However, for polynomial maps a characterization in terms of the polynomial coefficients is desirable and more applicable than Hadamard’s abstract characterization. The authors tackle this objective by giving sufficient criteria for the properness of \(F\) in terms of the coefficients of \(\|F\|_2^2\), making use of the Newton Polytope of \(\|F\|_2^2\) and the fact that \(\|F\|_2^2\) is a Sums-of-Squares polynomial. Key ingredients are the notions of gem regularity and circuit numbers.
In previous work [SIAM J. Optim. 25, No. 3, 1542–1570 (2015; Zbl 1322.90092)], the authors exhibited sufficient and necessary conditions for the properness of \(F\) that are being used in this article. Thus the present work focuses on the discussion of a parameterized family of illustrative examples. There, a small gap in between the necessary and sufficient criteria for properness manifests: The properness of \(F\) is invariant under linear transformations \(F\mapsto F\circ A\) (with \(A \in \text{GL}_n(\mathbb{R})\)), but the given sufficient criterion is not. The authors treat one example where direct application of their criterion fails and give an explicit linear map \(A\) that makes it applicable.
Reviewer: Alexander Taveira Blomenhofer (Konstanz)
MSC:
| 14P05 | Real algebraic sets |
| 26B10 | Implicit function theorems, Jacobians, transformations with several variables |
| 26C05 | Real polynomials: analytic properties, etc. |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 14R15 | Jacobian problem |
Keywords:
Newton polytope; coercivity; global invertibility; real Jacobian conjecture; circuit numberReferences:
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