Abstract
We begin with a survey of the notion of atypical values of R. Thom. We give a theorem on the atypical values of complex polynomials which generalizes a theorem of Broughton (On the topology of polynomial hypersurfaces in singularities, Part 1, 167–178, proceedings of symposia in pure mathematics, vol 40. American Mathematicaol Society, Providence, 1983) and Broughton (Milnor numbers and the topology of polynomial hypersurfaces. Invent Math 92:217–241, 1988). For this purpose we introduce the notion of atypical value from infinity. Our proofs are geometrical. We end with some open problems to understand the difference between tame polynomials at infinity and polynomials without atypical values from infinity.
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References
Bertini, E.: Introduction to the Projective Geometry of Hyperspaces. Messina, (1923)
Bertini, E.: Algebraic surfaces. Proc. Steklov Inst. Math. (Trudy Mat. Inst. Steklov 75 (1965)). 1967, 75 (1967)
Broughton, S.A.: On the Topology of Polynomial Hypersurfaces in Singularities, Part 1, 167–178, Proceedings of Symposia in Pure Mathematics, vol. 40. American Mathematical Society, Providence (1983)
Broughton, S.A.: Milnor numbers and the topology of polynomial hypersurfaces. Invent. Math. 92, 217–241 (1988)
Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscr. Math. 2, 103–161 (1970)
Hà, V.H., Lê, D.T.: Sur la topologie des polynômes complexes. Acta Math. Vietnam 9(1), 21–32 (1984)
Harris, J.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 133, p. xx+328. Springer, New York (1995)
Hatcher, A.: Algebraic Topology, p. xii+544. Cambridge University Press, Cambridge (2002)
Houzel, C.: Géométrie analytique locale I, in Sém. Cartan 1960/1961, Exposé 18, Pub. IHP
Lê, D.T.: Singularités isolées d’hypersurfaces. Acta Sc Vietnamicarum Tome 7, 24–33 (1971)
Mather, J.: Notes on topological stability. Bull. Am. Math. Soc. 49, 475–506 (2012)
Milnor, J.: Morse Theory (Based on Lecture Notes by M. Spivak and R. Wells), Annals of Mathematics Studies 51, p. 153. Princeton University Press, Princeton (1963)
Milnor, J.: Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies 61, p. 122. Princeton University Press, Princeton (1968)
Palamodov, V.P.: The multiplicity of a holomorphic transformation (in Russian). Funkc. Anal. i Priložen 1, 54–65 (1967)
Spanier, E.H.: Algebraic Topology, p. xiv+528. McGraw-Hill Book Co., New York (1966)
Strøm, A.: Note on cofibrations. Math. Scand. 19, 11–14 (1966)
Strøm, A.: Note on cofibrations. II. Math. Scand. 22, 130–142 (1968)
Tibǎr, M.: Polynomials and Vanishing Cycles Cambridge Tracts in Mathematics, vol. 170, p. xii + 253. Cambridge University Press, Cambridge (2007)
Verdier, J.L.: Stratifications de Whitney et théorème de Bertini–Sard. Invent. Math. 36, 295–312 (1976)
Whitney, H.: Elementary structure of real algebraic varieties. Ann. Math. 66, 545–556 (1957)
Whitney, H.: Tangents to an analytic variety. Ann. Math. 81, 496–549 (1965)
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Work partially supported by DGICYT Grant MTM2015-64013-P.
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Lê, D.T., Nuño Ballesteros, J.J. A remark on the topology of complex polynomial functions. RACSAM 113, 3977–3994 (2019). https://doi.org/10.1007/s13398-018-0611-z
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DOI: https://doi.org/10.1007/s13398-018-0611-z