Computing the equisingularity type of a pseudo-irreducible polynomial
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Computing the equisingularity type of a pseudo-irreducible polynomial Adrien Poteaux1 · Martin Weimann2,3 Received: 9 November 2019 / Revised: 15 June 2020 / Accepted: 24 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over C, this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation. Keywords Polynomials · Plane curves · Equisingularity · Pseudo-irreduibility · Approximate roots Mathematics Subject Classification 14Q20 · 12Y05 · 13P05 · 68W30
1 Introduction Equisingularity is the main notion of equivalence for germs of plane curves. It was developed in the 60’s by Zariski over algebraically closed fields of characteristic zero in [29–31] and generalised in arbitrary characteristic by Campillo [2]. This concept is of particular importance as for complex curves, it agrees with the topological equivalence class [29]. As illustrated by an extensive literature (see e.g. the book [9] and the references therein), equisingularity plays nowadays an important role in various active fields of singularity theory (resolution, equinormalisable deformation, moduli
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Adrien Poteaux [email protected] Martin Weimann [email protected]
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CRIStAL, CNRS UMR 9189, Université de Lille, Bâtiment Esprit, 59655 Villeneuve d’Ascq, France
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LMNO, CNRS UMR 6139, Université de Caen-Normandie, BP 5186, 14032 Caen Cedex, France
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Present Address: GAATI, EA 3893, Université de Polynésie Française, BP 6570, 98702 Faa’a, Polynésie Française
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A. Poteaux, M. Weimann
problems, analytic classification, etc). It is thus an important issue of computer algebra to design efficient algorithms for computing the equisingularity type of a singularity. This paper is dedicated to characterise a family of reduced germs of plane curves, containing irreducible ones, for which this task can be achieved in quasi-linear time with respect to the discriminant valuation of a Weierstrass equation. Main result. We say that two germs of reduced plane curves are equisingular if there is a one-to-one correspondence between their branches which preserves the characteristic exponents and the pairwise intersection multiplicities (see e.g. [2, 3, 27] for other equivalent definitions). This equivalence relation leads to the notion of equisingularity type of a singularity. In this paper, we consider a square-free Weierstrass polynomial F ∈ K[[x]][y] of degree d, with K a perfect field of characteristic zero or greater than d 1 . Under such an assumption, the Puiseux series of F are well defined and allow to determine the equisingularity type of the germ (F, 0) (the case of small characteristic requires Hamburger–Noether expansions [2]). In particular, it follows from [22] that we
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