# zbMATH — the first resource for mathematics

Asymptotic values of polynomial mappings of the real plane. (English) Zbl 1104.14022
Summary: Using algebraically constructible functions we give a generically effective method to detect asymptotic values of polynomial mappings with finite fibers defined on the real plane. By asymptotic variety we mean the set of points at which the polynomial mapping fails to be proper.
##### MSC:
 14P25 Topology of real algebraic varieties 26C99 Polynomials, rational functions in real analysis 30C10 Polynomials and rational functions of one complex variable
Full Text:
##### References:
 [1] Basu, S.; Pollack, R.; Roy, M.-F., Algorithms in real algebraic geometry, (2003), Springer Berlin [2] Coste, M.; Kurdyka, K., Le discriminant d’un morphisme de variétés algébriques réelles, Topology, 37, 2, 393-399, (1998) · Zbl 0942.14031 [3] Hermite, C., Sur l’extension du théorème de M. Sturm à un système d’équation simultanées, Oeuvres de charles Hermite, 3, 1-34, (1969) [4] Jelonek, Z., The set of points at which a polynomial map is not proper, Ann. polon. math., LVIII, 3, 259-266, (1993) · Zbl 0806.14009 [5] Jelonek, Z., Geometry of real polynomial mappings, Math. Z., 239, 321-333, (2002) · Zbl 0997.14017 [6] Jelonek, Z.; Kurdyka, K., On asymptotic critical values of a complex polynomial, J. reine angew. math., 565, 1-11, (2003) · Zbl 1047.32019 [7] McCrory, C.; Parusiński, A., Algebraically constructible functions, Ann. scient. éc. norm. sup $$4^{\operatorname{e}}$$ série, 30, 527-552, (1997) · Zbl 0913.14018 [8] McCrory, C.; Parusiński, A., Complex monodromy and the topology of real algebraic sets, Comp. math., 106, 527-552, (1997) [9] Parusiński, A.; Szafraniec, Z., Algebraically constructible functions and signs of polynomials, Manuscripta math., 93, 443-456, (1997) · Zbl 0913.14019 [10] Stasica, A., An effective description of the jelonek set, J. pure appl. algebra, 169, 2-3, 321-326, (2002) · Zbl 0999.14023 [11] Viro, O.Y., Some integral calculus based on Euler characteristic, (), 127-138
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.