Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials

Charles N. Delzell

doi:10.5802/afst.1274

Charles N. Delzell ¹

¹ Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 37-56.

Abstract
Abstract

Let $h : ℝ^{n} \to ℝ$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form ${sup}_{i} {inf}_{j} f_{i j}$ , for some finite collection of polynomials $f_{i j} \in ℝ [x_{1}, ..., x_{n}]$ . (A simple example is $h (x_{1}) = | x_{1} | = sup {x_{1}, - x_{1}}$ .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n \leq 2$ ; it remains open for $n \geq 3$ . In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers; in this version, we restrict to the positive orthant, where each $x_{i} > 0$ . As before, our methods work only for $n \leq 2$ .

En 1984, L. Mahé, et indépendammant G. Efroymson, ont prouvé le cas où $n \leq 2$ de la conjecture de Pierce-Birkhoff (1956) : une fonction $h : ℝ^{n} \to ℝ$ continue polynomiale par morceaux peut s’écrire comme ${sup}_{i} {inf}_{j} f_{i j}$ , pour une collection finie de polynômes $f_{i j} \in ℝ [x_{1}, ..., x_{n}]$ . (Un exemple simple est $h (x_{1}) = | x_{1} | = sup {x_{1}, - x_{1}}$ .) La conjecture reste ouverte pour $n \geq 3$ . Dans cet article, nous prouvons (encore pour $n \leq 2$ ) un résultat analogue pour « polynômes généralisés », où les exposants peuvent être des nombres réels arbitraires, et non pas seulement des nombres naturels ; dans cette version, nous limitons le domaine à l’orthant positif, où chaque $x_{i} > 0$ .

MR

DOI: 10.5802/afst.1274

Author's affiliations:

Charles N. Delzell ¹

¹ Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA

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     title = {Extension of the {Two-Variable} {Pierce-Birkhoff} conjecture to generalized polynomials},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Charles N. Delzell. Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 37-56. doi : 10.5802/afst.1274. https://afst.centre-mersenne.org/articles/10.5802/afst.1274/

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