Publication:
On the Set of Points at Infinity of a Polynomial Image of Rn

Loading...
Thumbnail Image
Full text at PDC
Publication Date
2014
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
In this work we prove that the set of points at infinity of a semialgebraic set that is the image of a polynomial map is connected. This result is no longer true in general if is a regular map. However, it still works for a large family of regular maps that we call quasi-polynomial maps.
Description
Keywords
Citation
[]BCR J. Bochnak, M. Coste, M.F. Roy: Real algebraic geometry. Ergeb. Math. 36, Springer-Verlag, Berlin: 1998. []F J.F. Fernando: On the one dimensional polynomial and regular images of Rn. J. Pure Appl.Algebra 218 (2014), no. 9, 1745–1753. []FG1 J.F. Fernando, J.M. Gamboa: Polynomial images of Rn. J. Pure Appl. Algebra 179 (2003), no. 3,241–254. []FG2 J.F. Fernando, J.M. Gamboa: Polynomial and regular images of Rn. Israel J. Math. 153 (2006),61–92. []FG3 J.F. Fernando, J.M. Gamboa: On the irreducible components of a semialgebraic set. Internat. J.Math. 23 (2012), no. 4, 1250031 (40 pages). []FGU1 J.F. Fernando, J.M. Gamboa, C. Ueno: On convex polyhedra as regular images of Rn. Proc. Lond.Math. Soc. (3) 103 (2011) 847–878. []FGU2 J.F. Fernando, J.M. Gamboa, C. Ueno: Properties of the boundary of polynomial and regular images of Rn. Mathematical contributions in honor of Juan Tarres, 159–178, Univ. Complut.Madrid, Fac. Mat., adrid: 2012. []FU1 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 http://imrn.oxfordjournals.org/content/early/2013/06/17/imrn.rnt112.full .pdf?keytype=ref&ijkey=PiOgJzspWYsUzGp []FU2 J.F. Fernando, C. Ueno: On the complements of 3-dimensional convex polyhedra as polynomial images of R3. Internat. J. Math. XXX (2014, accepted), no. X, XXX-XXX. []G J.M. Gamboa: Reelle algebraische Geometrie, June,10th ´ 16th (1990), Oberwolfach. []Ha J. Harris: Algebraic Geometry. A first course. raduate Texts in Mathematics, 133. Springer-Verlag, New York: 1992. []H1 R. Hartshorne: Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, 156 Springer-Verlag, erlin-New York: 1970. []H2 R. Hartshorne: Algebraic geometry. Graduate Texts in Mathematics, 52 Springer-Verlag, New York-Heidelberg:1977. []J1 Z. Jelonek: Testing sets for properness of olynomial mappings. Math. Ann. 315 (1999), no. 1,1–35. []J2 Z. Jelonek: Geometry of real polynomial mappings. Math. Z., 239 (2002), no. 2, 321–333. []JK Z. Jelonek, K. Kurdyka: Reaching generalized critical values of a polynomial. Math. Z. (2014, to appear), no. X, XXX-XXX. arXiv:1203.0539v2.pdf []M D. Mumford: Algebraic geometry. I. Complex projective varieties. Grundlehren der Mathematis-chen issenschaften, 221. Springer-Verlag, Berlin-New York: 1976. []P S. Pinchuk: A counterexample to the real Jacobian Conjecture, Math. Z. 217 (1994), 1–4. []Sh1 I.R. Shafarevich: Basic Algebraic Geometry I. Varieties in projective space. Second edition. Trans-lated from the 1988 Russian edition and with notes by iles Reid. Springer-Verlag, Berlin: 1994. []Sh2 I.R. Shafarevich: Basic Algebraic Geometry II. Schemes and complex manifolds. Second edition. Translated from the 1988 Russian edition by Miles Reid. Springer-erlag, Berlin: 1994. []S E.H. Spanier: Algebraic topology. Springer-Verlag, New York-Berlin: 1981. []U1 C. Ueno: A note on boundaries of open polynomial images of R2. Rev. Mat. Iberoam. 24 (2008),981-988. []U2 C. Ueno: On convex polygons and their complements as images of regular and polynomial maps of R2. J. Pure Appl. Algebra 216 (2012), no. 11, 2436–2448.
Collections