Polynomial and regular images of \(\mathbb R^n\). (English) Zbl 1213.14109
Summary: We obtain new necessary conditions for an\(n\)-dimensional semialgebraic subset of \(\mathbb R^n\) to be a polynomial image of \(\mathbb R^n\) . Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers.
MSC:
| 14P10 | Semialgebraic sets and related spaces |
References:
| [1] | J. Bochnak, M. Coste and M.-F. Roy,Géométrie algébrique réelle, Ergebnisse der Mathematik 12, Springer-Verlag, Berlin-Heidelberg-New York, 1987. |
| [2] | J. F. Fernando and J. M. Gamboa,Polynomial images of R n , Journal of Pure and Applied Algebra179 (2003), 241–254. · Zbl 1042.14035 · doi:10.1016/S0022-4049(02)00121-4 |
| [3] | Z. Jelonek,A geometry of polynomial transformations of the real plane, Bulletin of the Polish Academy of Sciences. Mathematics48 (2000), 57–62. · Zbl 0958.14042 |
| [4] | Z. Jelonek,Geometry of real polynomial mappings, Mathematische Zeitschrift239 (2002), 321–333. · Zbl 0997.14017 · doi:10.1007/s002090100298 |
| [5] | E. Nöther,Körper und Systeme rationaler Funktionen, Mathematische Annalen76 (1915), 161–196. · JFM 46.1442.08 · doi:10.1007/BF01458137 |
| [6] | A. Schinzel,Selected Topics on Polynomials, The University of Michigan Press, Ann Arbor, 1982. |
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