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Atypical Values and the Milnor Set of Real Polynomials in Two Variables

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Abstract

We give a new algorithmic method of detection of atypical values for 2-variables real polynomial functions with emphasis on the effectivity.

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Notes

  1. Note that this is not an order relation.

References

  • Araújo dos Santos, R.N., Chen, Y., Tibăr, M.: Singular open book structures from real mappings. Cent. Eur. J. Math. 11(5), 817–828 (2013)

  • Araújo dos Santos, R.N., Chen, Y., Tibăr, M.: Real polynomial maps and singular open books at infinity. Math. Scand. 118(1), 57–69 (2016)

  • Bochnak, J., Coste, M., Roy, M.F. Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36, Springer, Berlin (1998) (Translated from the 1987 French original, Revised by the authors)

  • Coste, M., de la Puente, M.J.: Atypical values at infinity of a polynomial function on the real plane: an erratum, and an algorithmic criterion. J. Pure Appl. Algebra 162(1), 23–35 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Dias, L.R.G., Tibăr, M.: Detecting bifurcation values at infinity of real polynomials. Math. Z. 279(1–2), 311–319 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Dias, L.R.G., Ruas, M.A.S., Tibăr, M.: Regularity at infinity of real mappings and a Morse–Sard theorem. J. Topol. 5(2), 323–340 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Dias, L.R.G., Tanabé, S., Tibăr, M.: Toward effective detection of the bifurcation locus of real polynomial maps. Found. Comput. Math. 17, 837–849 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Dias, L.R.G., Joiţa, C., Tibăr, M.: Atypical points at infinity and algorithmic detection of the bifurcation locus of real polynomials. Math. Z. 298(3–4), 1545–1558 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Hà, H.V., Lê, D.T.: Sur la topologie des polynomês complexes. Acta Math. Vietnam 9, 21–32 (1984)

    MathSciNet  MATH  Google Scholar 

  • Hà, H.V., Nguyên, L.A.: Atypical values at infinity of polynomial and rational functions on an algebraic surface in \( {\mathbb{R} }^n\). Acta Math. Vietnam 36(2), 537–553 (2011)

    MathSciNet  MATH  Google Scholar 

  • Joiţa, C., Tibăr, M.: Bifurcation values of families of real curves. Proc. Roy. Soc. Edinburgh Sect. A 147(6), 1233–1242 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Milnor, J. Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo,(1968)

  • Monsalve, G.E., Tibăr, M.: On the gradient index at infinity of real 2-variables polynomials, Manuscript

  • Parusiński, A.: On the bifurcation set of complex polynomial with isolated singularities at infinity. Compos. Math. 97, 369–384 (1995)

    MathSciNet  MATH  Google Scholar 

  • Siersma, D., Tibăr, M.: Singularities at infinity and their vanishing cycles. Duke. Math. J. 80(3), 771–783 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Suzuki, M.: Properiétés topologiques des polynômes des deux variables complexes. Math Soc. Jpn. 26, 241–257 (1974)

    MATH  Google Scholar 

  • Thom, R.: Ensembles et morphismes stratifiés. Bull. Am. Math. Soc. 75, 249–312 (1996)

    MATH  Google Scholar 

  • Tibăr, M.: Topology at infinity of polynomial mappings and Thom condition. Compos. Math. 111, 89–109 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Tibăr, M.: Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996), pp. xx, 249–264 (1999)

  • Tibăr, M.: Polynomials and vanishing cycles. Cambridge Tracts in Mathematics, vol. 170. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  • Tibăr, M., Zaharia, A.: Asymptotic behaviour of families of real curves. Manuscripta Math. 99(3), 383–393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini–Sard, Invent. Math. 36, 295–312

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Correspondence to Gabriel E. Monsalve.

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This work was supported by the Grant \(\#\)2019/24377-2 and \(\#\)2020/14111-2, São Paulo Research Foundation (FAPESP). The author thanks to Prof. Mihai Tibăr for his guidance in this topic.

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Monsalve, G.E. Atypical Values and the Milnor Set of Real Polynomials in Two Variables. Bull Braz Math Soc, New Series 54, 13 (2023). https://doi.org/10.1007/s00574-022-00328-2

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  • DOI: https://doi.org/10.1007/s00574-022-00328-2

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