Real Algebraic and Analytic Geometry

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283. Andreas Fischer, Murray Marshall:
Extending piecewise polynomial functions in two variables.

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Submission: 2009, May 28.

We study the extendibility of piecewise polynomial functions defined on closed subsets of $\mathbb{R}^2$. The compact subsets on which every piecewise polynomial function is extensible can be characterized in terms of quasi-convexity if they are definable in an o-minimal expansion of the real field. Even the non compact 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 subset of $\mathbb{R}^2$ which is not extensible.

Mathematics Subject Classification (2000): 14P10, 03C64.

Keywords and Phrases: piecewise polynomial function, Pierce Birkhoff ring.

Full text, 11p.: dvi 54k, ps.gz 154k, pdf 192k.

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