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On the complements of 3-dimensional convex polyhedra as polynomial images of R3

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http://arxiv.org/pdf/1212.1815v3.pdf
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2014
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World Scientific
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Let K Rn be a convex polyhedron of dimension n. Denote S := RnK and let S be its closure. We prove that for n = 3 the semialgebraic sets S and S are polynomial images of 3. The former techniques cannot be extended in general to represent the semialgebraic sets S and S as polynomial images of n if n ≥ 4.
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Álgebra
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1201 Álgebra
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M. Berger: Geometry. I & II. Universitext. Springer-Verlag, Berlin: 1987. J. Bochnak, M. Coste, M.F. Roy: Real algebraic geometry. Ergeb. Math. 36. Springer-Verlag, Berlin: 1998. J.F. Fernando: On the one dimensional polynomial and regular images of Rn . J. Pure Appl. Algebra XXX (2014, accepted), no. X, XXX–XXX. J.F. Fernando, J.M. Gamboa: Polynomial images of Rn . J. Pure Appl. Algebra 179 (2003), no. 3, 241–254. J.F. Fernando, J.M. Gamboa: Polynomial and regular images of R n. Israel J. Math. 153 (2006), 61–92. J.F. Fernando, J.M. Gamboa, C. Ueno: On convex polyhedra as regular images of Rn. Proc. London Math.Soc. (3) 103 (2011), 847–878. J.F. Fernando, C. Ueno: On the set of points at infinity of a polynomial image of Rn. Preprint RAAG (2011).arXiv:1212.1811 J.F. Fernando, C. Ueno: On complements of convex polyhedra as polynomial and regular images of Rn. Int. Math. Res. Not. IMRN XXX (2013, accepted), no. X, XXX–XXX. J.M. Gamboa: Reelle Algebraische Geometrie, June, 10th − 16th (1990), Oberwolfach.[R] T.R. Rockafellar: Convex analysis. Princeton Mathematical Series, 28. Princeton University Press, Princeton,N.J: 1970. C. Ueno: A note on boundaries of open polynomial images of R 2. Rev. Mat. Iberoam. 24 (2008), no. 3,981-988. C. Ueno: On convex polygons and their complements as images of regular and polynomial maps of R2. J.Pure Appl. Algebra 216, no. 11, 2436–2448. C. Ueno: Unbounded convex polygons as polynomial images of the plane. Preprint RAAG (2013).arXiv:1307.8341 G.M. Ziegler: Lectures on Polytopes. Graduate Texts in Mathematics 152. Springer-Verlag, New-York:1995
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https://hdl.handle.net/20.500.14352/34962
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