no. 2
Extending piecewise polynomial functions in two variables
Fischer, Andreas1 ; Marshall, Murray2 idref
1Fields Institute, Toronto, Canada, current: Comenius Gymnasium Datteln, Südring150, 45711 Datteln, Germany
2University of Saskatchewan, Department of Mathematics & Statistics, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 2, pp. 253-268

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 .

DOI : 10.5802/afst.1372

Fischer, Andreas  1   ; Marshall, Murray  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 
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     title = {Extending piecewise polynomial functions in two variables},
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     pages = {253--268},
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     volume = {Ser. 6, 22},
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Fischer, Andreas; Marshall, Murray. Extending piecewise polynomial functions in two variables. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 2, pp. 253-268. doi: 10.5802/afst.1372

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

[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). | Zbl | MR

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

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

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

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

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

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

[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). | Zbl | MR

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

[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). | Zbl | MR

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

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

[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). | Zbl | MR

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

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