Domains of unicity
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VILMOS TOTIK∗ In honor of Lawrence Zalcman Abstract. The Gale–Nikaido theorem claims that if the Jacobian of a mapping F is a P-matrix at every point of K and K is a closed rectangular region in Rn , then F is globally univalent on K. Under the more severe condition that the (symmetric part of the) Jacobian is positive definite on K, the same conclusion is valid on any closed convex set K. In this paper it is shown that the closed rectangular regions are the only ones for which the Gale–Nikaido theorem is true. In a similar fashion, it is shown that the positive definiteness of the Jacobian implies unicity only on (closed) convex sets.
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Introduction
It is well known if F = (Fi (x1 , . . . , xn ))ni=1 is a differentiable mapping from a subset K of Rn into Rn and if the Jacobian (∂Fi /∂xj ) of F does not vanish at a point, then F is univalent (1-to-1) in a neighborhood of that point. Global univalence is more subtle, and the mere vanishing of the Jacobian at every point of K is not sufficient. Gale and Nikaido [4] proved in 1965 that if the Jacobian of F is a P-matrix at every point of K (meaning that all of its principal minors are positive) and K is a closed rectangular region1 [ai , bi ] (ai < bi for all i), then F is injective. Under the more severe condition that the (symmetric part of the) Jacobian is positive definite on K, the same conclusion is valid on any convex set K; see [3], [4] and [5]. The problem if the Gale–Nikaido theorem is true on any convex set has been mentioned several times in the book [6], but counterexamples were given later in [1] and [7]. In this note we address the question on what domains are the aforementioned two unicity theorems true. We are going to show that the closed rectangular regions are the only ones for which the Gale–Nikaido theorem is true, which makes that ∗
Supported by NSF grant DMS 1564541.
1 This terminology follows the original paper [4].
parallelepiped.”
A more correct notion would be “closed rectangular
397 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0117-4
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V. TOTIK
result quite a peculiar one. In a similar fashion, we shall show that positive definiteness of the Jacobian implies unicity only on (closed) convex sets. Let K ⊂ Rn , n ≥ 2, be a compact set. In what follows we shall consider continuously differentiable maps F from K to Rn , and in order not to worry about the notion of the partial derivatives at arbitrary points of K, we shall assume without mentioning that F is defined on a neighborhood of K. Recall that a not necessarily symmetric (real) square matrix is called a P-matrix if all of its principal submatrices (obtained by deleting some rows and the corresponding columns) have positive determinant. See [2, Section 5.5] or [4] for properties of P-matrices. Recall also that a symmetric square matrix A is positive definite if x∗ Ax > 0 for all non-zero vectors x, where ·∗ denotes transposition. By Sylvester’s criterion this happens if and only if all leading principal submatrices of A have positive determinan
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