Extending piecewise polynomial functions in two variables

Andreas Fischer; Murray Marshall

doi:10.5802/afst.1372

Andreas Fischer ¹; Murray Marshall ²

¹ Fields Institute, Toronto, Canada, current: Comenius Gymnasium Datteln, Südring150, 45711 Datteln, Germany
² University of Saskatchewan, Department of Mathematics & Statistics, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 2, pp. 253-268.

Abstract
Abstract

We study the extensibility of piecewise polynomial functions defined on closed subsets of $ℝ^{2}$ to all of $ℝ^{2}$ . The compact subsets of $ℝ^{2}$ on which every piecewise polynomial function is extensible to $ℝ^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$ . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial function defined on a compact, convex, but undefinable subset of $ℝ^{2}$ which is not extensible to $ℝ^{2}$ .

Nous étudions le prolongement des fonctions polynômes par morceaux définies sur des sous-ensembles fermés de $ℝ^{2}$ à tout $ℝ^{2}$ . Les sous-ensembles compacts de $ℝ^{2}$ sur lesquels chaque fonction polynôme par morceaux est prolongeable à $ℝ^{2}$ peuvent être caractérisés en termes de quasi-convexité locale si ils sont définissables dans une expansion o-minimale de $ℝ$ . Même les sous-ensembles non compacts fermés définissables peuvent être caractérisés si les germes de fonctions semi-algébriques à l’infini sont denses dans le corps de Hardy des germes définissables. Nous présentons également une fonction polynôme par morceaux définie sur un sous-ensemble compact, convexe, mais indéfinissable de $ℝ^{2}$ , et qui n’est pas prolongeable à $ℝ^{2}$ .

Zbl

DOI: 10.5802/afst.1372

Author's affiliations:

Andreas Fischer ¹; Murray Marshall ²

¹ Fields Institute, Toronto, Canada, current: Comenius Gymnasium Datteln, Südring150, 45711 Datteln, Germany
² University of Saskatchewan, Department of Mathematics & Statistics, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

@article{AFST_2013_6_22_2_253_0,
     author = {Andreas Fischer and Murray Marshall},
     title = {Extending piecewise polynomial functions in two variables},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {253--268},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 22},
     number = {2},
     year = {2013},
     doi = {10.5802/afst.1372},
     zbl = {1279.14069},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1372/}
}

TY  - JOUR
AU  - Andreas Fischer
AU  - Murray Marshall
TI  - Extending piecewise polynomial functions in two variables
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2013
SP  - 253
EP  - 268
VL  - 22
IS  - 2
PB  - Université Paul Sabatier, Institut de Mathématiques
PP  - Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1372/
DO  - 10.5802/afst.1372
LA  - en
ID  - AFST_2013_6_22_2_253_0
ER  -

%0 Journal Article
%A Andreas Fischer
%A Murray Marshall
%T Extending piecewise polynomial functions in two variables
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2013
%P 253-268
%V 22
%N 2
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1372/
%R 10.5802/afst.1372
%G en
%F AFST_2013_6_22_2_253_0

Andreas Fischer; Murray Marshall. Extending piecewise polynomial functions in two variables. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 2, pp. 253-268. doi : 10.5802/afst.1372. https://afst.centre-mersenne.org/articles/10.5802/afst.1372/

References
Cited by

[1] Bochnak (J.), Coste (M.), Roy (M.-F.).— Géométrie algébrique réelle, Ergeb. Math. 12, Springer (1987). Real algebraic geometry, Ergeb. Math. 36, Springer (1998). | MR | Zbl

[2] Delzell (C.N.).— On the Pierce-Birkhoff conjecture over ordered fields. Quadratic forms and real algebraic geometry (Corvallis, OR, 1986). Rocky Mountain J. Math. 19, no. 3, p. 651-668 (1989). | MR | Zbl

[3] Delzell (C.N.).— Continuous, piecewise-polynomial functions which solve Hilbert’s 17th problem. J. Reine Angew. Math. 440, p. 157-173 (1993). | MR | Zbl

[4] Denef (J.), van den Dries (L.).— $p$ -adic and real subanalytic sets. Ann. of Math. (2) 128, no. 1, p. 79-138 (1988). | MR | Zbl

[5] van den Dries (L.), Miller (C.).— Geometric categories and o-minimal structures. Duke Math. J. 84, no. 2, p. 497-540 (1996). | MR | Zbl

[6] Fischer (A.).— O-minimal $Λ^{m}$ -regular Stratification. Ann. Pure Appl. Logic, 147, no. 1-2, p. 101-112 (2007). | MR | Zbl

[7] Fischer (A.).— O-minimal analytic separation of sets in dimension 2. Ann. Pure Appl. Logic, 157, (2009) no. 2-3, 130-138. | MR | Zbl

[8] Henriksen (M.), Isbell (J.R.).— Lattice-ordered rings and function rings. Pacific J. Math. 12, p. 533-565 (1962). | MR | Zbl

[9] Kurdyka (K.).— On a subanalytic stratification satisfying a Whitney-Property with exponent 1. Proceeding Conference Real Algebraic Geometry - Rennes 1991, Springer LNM 1524, p. 316-322 (1992). | MR | Zbl

[10] Madden (J.J.).— Pierce-Birkhoff rings. Arch. Math. (Basel) 53, no. 6, p. 565-570 (1989). | MR | Zbl

[11] Mahé (L.).— On the Pierce-Birkhoff conjecture. Ordered fields and real algebraic geometry (Boulder, Colo., 1983). Rocky Mountain J. Math. 14, no. 4, p. 983-985 (1984). | MR | Zbl

[12] Mahé (L.).— On the Pierce-Birkhoff conjecture in three variables. J. Pure Appl. Algebra 211, no. 2, p. 459-470 (2007). | MR | Zbl

[13] Marshall (M.).— The Pierce-Birkhoff conjecture for curves. Canad. J. Math. 44, no. 6, p. 1262-1271 (1992). | MR | Zbl

[14] Rolin (J.-P.), Speissegger (P.), Wilkie (A.J.).— Quasianalytic Denjoy-Carleman classes and o-minimality. J. Amer. Math. Soc. 16, no. 4, p. 751-777 (2003). | MR | Zbl

[15] Schwartz (N.).— Piecewise polynomial functions. Ordered algebraic structures (Gainesville, FL, 1991), Kluwer Acad. Publ., Dordrecht, p. 169-202 (1993). | MR | Zbl

Cited by Sources: