## Publication: On Complements of Convex Polyhedra as Polynomial
and Regular Images of Rn

Loading...

##### Official URL

##### Full text at PDC

##### Publication Date

2013

##### Advisors (or tutors)

##### Editors

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

y Oxford University Press

##### Abstract

In this work we prove constructively that the complement Rn \ K of a convex polyhedron K ⊂ Rn and the complement Rn \ Int(K) of its interior are regular images of Rn. If K is
moreover bounded, we can assure that Rn \ K and Rn \ Int(K) are also polynomial images of Rn. The construction of such regular and polynomial maps is done by double induction
on the number of facets (faces of maximal dimension) and the dimension of K; the careful placing (first and second trimming positions) of the involved convex polyhedra which
appear in each inductive step has interest by its own and it is the crucial part of our technique.

##### Description

##### UCM subjects

##### Unesco subjects

##### Keywords

##### Citation

[1] Berger, M. “Geometry. I y II.” Universitext, Berlin: Springer, 1987.
[2] Bochnak, J., M. Coste, and M. F. Roy. Real Algebraic Geometry, Ergebnisse der Mathematik 36. Berlin: Springer, 1998.
[3] Fernando, J. F. “On the one dimensional polynomial and regular images of Rn.” (2013): preprint RAAG.
[4] Fernando, J. F. and J. M. Gamboa. “Polynomial images of Rn.” Journal of Pure and Applied Algebra 179, no. 3 (2003): 241–54.
[5] Fernando, J. F. and J. M. Gamboa. “Polynomial and regular images of Rn.” Israel Journal of Mathematics 153 (2006): 61–92.
[6] Fernando, J. F., J. M. Gamboa, and C. Ueno. “On convex polyhedra as regular images of Rn.” Proceedings of the London Mathematical Society (3) 103, no. 5 (2011): 847–78.
[7] Fernando, J. F. and C. Ueno. “On the set of points at infinity of a polynomial image of Rn.”(2012): preprint RAAG.
[8] Fernando, J. F. and C. Ueno. “On the complements of 3-dimensional convex polyhedra as polynomial images of R3 and its relation with sectional projections of convex polyhedra.” (2012): preprint RAAG.
[9] Gamboa, J. M. Reelle Algebraische Geometrie, June, 10–16, Oberwolfach, 1990.
[10] Heintz, J., T. Recio, and M. F. Roy. “Algorithms in Real Algebraic Geometry and Applications to Computational Geometry.” In Discrete and Computational Geometry: Papers from the DIMACS Special Year, edited by J.E. Goodman, R.D. Pollack, and W.L. Steiger. Series in Discrete Mathematics and Theoretical Computer Science 6. AMS-ACM, 1991.
[11] Nie, J., J. Demmel, and B. Sturmfels. “Minimizing polynomials via sum of squares over the gradient ideal.” Mathematical Programming 106, no. 3 (2006): Ser. A, 587–606.
[12] Parrilo, P. A. and B. Sturmfels, “Bernd Minimizing polynomial Functions.” Algorithmic and Quantitative Real Algebraic Geometry, 83–99. Piscataway, NJ, 2001. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. 60, Providence, RI: American Mathematical Society, 2003.
[13] Rockafellar, T. R. Convex Analysis, Princeton Mathematical Series 28. Princeton, NJ: Princeton University Press, 1970.
[14] Schweighofer, M. “Global optimization of polynomials using gradient tentacles and sums of squares.” SIAM Journal on Optimization 17, no. 3 (2006): 920–42
[15] Stengle, G. “A nullstellensatz and a positivstellensatz in semialgebraic geometry.” Mathematische Annalen 207 (1974): 87–97.
[16] Ueno, C. “A note on boundaries of open polynomial images of R2.” Revista Matematica Iberoamericana 24, no. 3 (2008): 981–8.
[17] Ueno, C. “On convex polygons and their complements as images of regular and polynomial maps of R2.” Journal of Pure and Applied Algebra 216, no. 11 (2012): 2436–48.
[18] Ueno, C. “Convex polygons via polynomial origami.” (2012): preprint RAAG.
[19] Vui, H. H. and P. T. Son. “Global optimization of polynomials using the truncated tangency variety and sums of squares.” SIAM Journal on Optimization 19, no. 2 (2008): 41–51.