Cubics as Tools to Study the Topology of M-Curves of Degree 9 in R <i>P</i>
                        <sup>2</sup>

Touzé, Séverine Fiedler-Le

doi:10.1112/s002461070200323x

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“…The proof is an improvement of the classical restriction method, with the addition of more recent tools: auxiliary pencils of rational cubics [4] and Orevkov's complex orientation formulas for an M -curve with deep nest [11]. The auxiliary cubics [2,3] and Orevkov's formulas [11,13] were already applied successfully to the study of M -curves of degree 9. The results in the present paper, as well as in [5], furnish evidence that the combination of these two tools may be fruitful.…”

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confidence: 99%

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M ‐curves of degree 9 with deep nests

Touzé

2009

Journal of the London Mathematical Society

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The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of given degree m. For m ≥ 8, one restricts the study to the case of the M -curves. For m = 9, the classification is still wide open. We say that an M -curve of degree 9 has a deep nest if it has a nest of depth 3. In the present paper, we prohibit 10 isotopy types with deep nest and no outer ovals.

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Section: Introductionmentioning

confidence: 99%

“…See e.g. [10], [7], [8], [9], [13], [17] for the constructions, and [5], [6], [10], [1], [2], [3], [4] [12], [14], [15], [17] for the restrictions.…”

Section: Introduction 1real and Complex Schemesmentioning

confidence: 99%

M ‐curves of degree 9 with deep nests

Touzé

2009

Journal of the London Mathematical Society

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“…Korchagin. Analysing available examples, he formulated [3] the following conjectures about the parity of the numbers α i in isotopy types of the form J α 0 1 α 1 . .…”

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confidence: 99%