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

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 (2010)

  • Volume: 19, Issue: S1, page 37-56
  • ISSN: 0240-2963

Abstract

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Let h : ℝ n → 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 ∈ 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 ≤ 2 ; it remains open for n ≥ 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 ≤ 2 .

How to cite

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Delzell, Charles N.. "Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 37-56. <http://eudml.org/doc/115901>.

@article{Delzell2010,
abstract = {Let $h:\mathbb\{R\}^n\rightarrow \mathbb\{R\}$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form $\sup _i\inf _jf_\{ij\}$, for some finite collection of polynomials $f_\{ij\}\in \mathbb\{R\}[x_1,\ldots ,x_n]$. (A simple example is $h(x_1)=|x_1|=\sup \lbrace x_1,-x_1\rbrace $.) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n\le 2$; it remains open for $n\ge 3$. 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 $x_i&gt;0$. As before, our methods work only for $n\le 2$.},
affiliation = {Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA},
author = {Delzell, Charles N.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {two-variable Pierce-Birkhoff conjecture; generalized polynomials},
language = {eng},
month = {4},
number = {S1},
pages = {37-56},
publisher = {Université Paul Sabatier, Toulouse},
title = {Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials},
url = {http://eudml.org/doc/115901},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Delzell, Charles N.
TI - Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 37
EP - 56
AB - Let $h:\mathbb{R}^n\rightarrow \mathbb{R}$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form $\sup _i\inf _jf_{ij}$, for some finite collection of polynomials $f_{ij}\in \mathbb{R}[x_1,\ldots ,x_n]$. (A simple example is $h(x_1)=|x_1|=\sup \lbrace x_1,-x_1\rbrace $.) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n\le 2$; it remains open for $n\ge 3$. 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 $x_i&gt;0$. As before, our methods work only for $n\le 2$.
LA - eng
KW - two-variable Pierce-Birkhoff conjecture; generalized polynomials
UR - http://eudml.org/doc/115901
ER -

References

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  4. C. Delzell, “Suprema of infima of generalized rational functions,” Abstract of a talk presented in: “Workshop: Real algebra, quadratic forms and model theory; algorithms and applications, November 2–9, 2005,” held during and as part of the Special Trimester on Real Geometry (September–December 2005), Centre Emile Borel, Institut Henri Poincaré, Paris. (Abstract published in the Workshop program: http://perso.univ-rennes1.fr/michel.coste/Borel/w1prog.html; see also http://www.ihp.jussieu.fr/ceb/Trimestres/T05-3/C1/index.html.) 
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  7. A.W. Hager and D.G. Johnson, Some comments and examples on generation of (hyper-)archimedean ℓ -groups and f -rings, Annales Faculté Sciences Toulouse, in press. Zbl1228.06008
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  10. L. Mahé, On the Pierce-Birkhoff conjecture, Rocky Mountain J. Math. 14 (1984), 983–5. Zbl0578.41008MR773148
  11. Chris Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), 79–94. Zbl0823.03018MR1278550
  12. Jesús Ruiz, The Basic Theory of Power Series, Advanced Lectures in Mathematics, Vieweg, 1003. MR1234937
  13. C. Sturm, “Extrait d’un Mémoire de M. Sturm, presenté à l’Académie des sciences, dans un séance du I er juin 1829,” Bulletin des Sciences Mathématiques, Physiques, et Chimiques, 1 re Section du Bulletin Universel, publié sous les auspices de Monseigneur le Dauphin, par la Société pour la Propagation des Connaissances Scientifiques et Industrielles, et sous la Direction de M. Le Baron de Férussac, Paris, Vol. 11 (1829), article # 272, pp. 422–5. 

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