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Atypical points at infinity and algorithmic detection of the bifurcation locus of real polynomials

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Abstract

We show that the variation of the topology at infinity of a two-variable polynomial function is localisable at a finite number of “atypical points” at infinity. We construct an effective algorithm with low complexity in order to detect sharply the bifurcation values of the polynomial function.

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Notes

  1. If instead of \(\hbox {C}^{\infty }\) we ask only “\(\hbox {C}^0\) trivial fibration” in the definition of typical value, then simple examples like \(f(x,y) = x^{3}+y^{2}\) show that a singular fibre may be typical (the fibre \(f^{-1}(0)\) in this case).

  2. In [5], the authors do not use the notion of “splitting” but something else called “cleaving”. They use parametrisations of affine curves without any truncation (whereas examples are worked out with some truncations), and effectivity is no concern either.

  3. The papers [22, 25, 28] present more general statements in \(n\ge 2\) complex variables, all of which amount to the same in case \(n=2\).

  4. Let us remark here that the sets \(f(\mathrm{Sing}f)\) and \({{\mathcal {A}}}_{f}\) may be disjoint, see Example 2.5.

  5. See also [28, Definition 1.1.5], [15, Theorem 4.7], [6, Definition 2.1], [8, Definition 2.1], [12, Page 1], [13, Page 1].

  6. The space \({{\mathbb {X}}}\) is not necessarily the closure in \({{\mathbb {P}}}^2\times {{\mathbb {R}}}\) of the graph of f like it is the case over \({{\mathbb {C}}}\), see [28, Note 1.1.3], for instance in the example \(f(x,y)=x^4 + y^2\).

  7. One may compare our definition of vanishing at \(\lambda \) to the contrary of “no vanishing at \(\lambda \)” as stated in [29, 15, Definition 3.1]. Similarly, the precise contrary of what we call here splitting at \(\lambda \) compares to the notion of “strong non-splitting” of [15, Definition 4.3]. The first part of the paper [15], as well as the whole paper [29], use a slightly weaker notion of “splitting” than in our Definition 2.7. The difference is that the non-splitting of [29] did not include the behaviour of compact components whereas our Definition 2.7 does. See [29, Example 3.2] where a circle component “splits” into a line component.

  8. Equivalently: (2’). there exist j and a continuous family of analytic paths \(\gamma _t:[0,1]\rightarrow X^{j}_t\) such that the limit set \(\lim _{t\rightarrow \lambda } \mathrm{Im}\gamma _t\) is not connected.

  9. See e.g. [19].

  10. As in [28, Definition 3.1.8].

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Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal de Uberlândia, Av. João Naves de Ávila 2121, 1F-153, Uberlândia, CEP: 38408-100, Brazil

    Luis Renato G. Dias

  2. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania

    Cezar Joiţa

  3. Laboratoire Paul Painlevé, University Lille, CNRS, UMR 8524, 59000, Lille, France

    Mihai Tibăr

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  1. Luis Renato G. Dias
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  2. Cezar Joiţa
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  3. Mihai Tibăr
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Correspondence to Mihai Tibăr.

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LRGD and MT acknowledge partial support by the Math-AmSud Grant 18-MATH-08. LRGD acknowledges the CNPq-Brazil Grants 401251/2016-0 and 304163/2017-1. CJ acknowledges the CNCS GrantPN-III-P4-ID-PCE-2016-0330. The authors acknowledge support by the Labex CEMPI (ANR-11-LABX-0007-01).

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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, 1545–1558 (2021). https://doi.org/10.1007/s00209-020-02662-x

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