Extending piecewise polynomial functions in two variables
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 2, p. 253-268

We study the extensibility of piecewise polynomial functions defined on closed subsets of 2 to all of 2 . The compact subsets of 2 on which every piecewise polynomial function is extensible to 2 can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of . 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 2 which is not extensible to 2 .

Nous étudions le prolongement des fonctions polynômes par morceaux définies sur des sous-ensembles fermés de 2 à tout 2 . Les sous-ensembles compacts de 2 sur lesquels chaque fonction polynôme par morceaux est prolongeable à 2 peuvent être caractérisés en termes de quasi-convexité locale si ils sont définissables dans une expansion o-minimale de . 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 2 , et qui n’est pas prolongeable à 2 .

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

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