Abstract
Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over \(\mathbb {C}\), 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.
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Notes
Our results still hold under the weaker assumption that the characteristic of \(\mathbb {K}\) does not divide \({d}\).
As usual, the notation \({{\mathcal {O}}}^{\sim}()\) hides logarithmic factors
The terminology pseudo-irreducible is also used in [14] to design polynomials which cannot be factored into comaximal polynomials. These two notions are not related to each other.
In the sequel, we rather use the terminology balanced and give an alternative definition of pseudo-irreducibility based on a Newton–Puiseux type algorithm. Both notions agree from Theorem 2.
Note that F being Weierstrass, we have \({d}\le {\delta}\) and \({\delta}\log ({d})\in {{\mathcal {O}}}^{\sim}({\delta})\).
When F is Weierstrass, we have \({d}\le {\delta}\). Otherwise, we might have \({d}\notin {{\mathcal {O}}}^{\sim}({\delta})\).
We can extend this definition to non-Weierstrass polynomials, see Sect. 5.3.
We may allow \(m_1=B_1=0\) when considering non Weierstrass polynomials, see Sect. 5.3.
In [23, Prop.12], the condition \(P_k(0)\in \mathbb {K}_k^\times \) is imposed even if \(q_k=1\), but this has no impact from a complexity point of view.
References
Abhyankar, S.: Irreducibility criterion for germs of analytic functions of two complex variables. Adv. Math. 35, 190–257 (1989)
Campillo, A.: Algebroid Curves in Positive Characteristic, volume 378 of LNCS. Springer, Berlin (1980)
Casas-Alvero, E.: Plane Curve Singularities, Volume 276 of LMS Lecture Notes. Cambridge University Press, Cambridge (2000)
Della Dora, J., Dicrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: EUROCAL 85, LNCS 204. Springer (1985)
Duval, D.: Rational Puiseux expansions. Compos. Math. 70(2), 119–154 (1989)
Garciá Barroso, E.R.: Invariants des singularités de courbes planes et courbure des fibres de Milnor. Ph.D. Tesis, ftp://tesis.bbtk.ull.es/ccppytec/cp16.pdf (1995)
Gathen, J.V.Z., Gerhard, J.: Modern Computer Algebra, 3rd edn. Cambridge University Press, New York (2013)
Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel (1994)
Greuel, G.-M., Lossen, C., Shustin, E.I.: Introduction to Singularities and Deformation Monographs in Mathematics. Springer, Berlin (2007)
Hodorog, M., Mourrain, B., Schicho, J.: A symbolic-numeric algorithm for computing the Alexander polynomial of a plane curve singularity. In: Proceedings of the 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, pp. 21–28 (2010)
Hodorog, M., Mourrain, B., Schicho, J.: An adapted version of the Bentley–Ottmann algorithm for invariants of plane curve singularities. In: Proceedings of 11th International Conference on Computational Science and Its Applications, volume 6784, pp. 121–131. Springer (2011)
Hodorog, M., Schicho, J.: A regularization method for computing approximate invariants of plane curves singularities. In: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation, pp. 44–53. SNC (2011)
Hoeven, J.V.D., Lecerf, G.: Accelerated Tower Arithmetic. Preprint hal-01788403 (2018)
MacAdam, S., Swan, R.: Factorizations of monic polynomials. In: Rings, Modules, Algebras, and Abelian Groups, volume 236 of Lecture Notes in Pure and Application Mathematics, pp. 411–424 (2004)
Merle, M.: Invariants polaires des courbes planes. Invent. Math. 41, 103–111 (1977)
Moroz, G., Schost, É.: A fast algorithm for computing the truncated resultant. In: ISSAC ’16: Proceedings of the Twenty-first International Symposium on Symbolic and Algebraic Computation, pp. 1–8, New York, NY, USA, 2016. ACM (2016)
Nguyen, T.-S., Pham, Hong-Duc, Hoang, P.-D.: Topological invariants of plane curve singularities: polar quotients and Lojasiewicz gradient exponents (2017). arXiv:1708.08295v1
Popescu-Pampu, P.: Approximate roots. Fields Inst. Commun. 33, 1–37 (2002)
Poteaux, A.: Calcul de développements de Puiseux et application au calcul de groupe de monodromie d’une courbe algébrique plane. Ph.D. thesis, Université de Limoges (2008)
Poteaux, A., Rybowicz, M.: Complexity bounds for the rational Newton–Puiseux algorithm over finite fields. Appl. Algebra Eng. Commun. Comput. 22, 187–217 (2011)
Poteaux, A., Rybowicz, M.: Improving complexity bounds for the computation of Puiseux series over finite fields. In: Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC’15, pp. 299–306, New York, NY, USA, 2015. ACM (2015)
Poteaux, A., Weimann, M.: Computing puiseux series : a fast divide and conquer algorithm, 2017. Preprint arXiv:1708.09067 (2017)
Poteaux, A., Weimann, M.: A quasi-linear irreducibility test in \({\mathbb{K}} [[x]][y]\), 2019. Preprint arXiv:1911.03551 (2019)
Poteaux, A., Weimann, M.: Using approximate roots for irreducibility and equi-singularity issues in \({\mathbb{K}}[[x]][y]\), 2019. Preprint arXiv:1904.00286v2 (2019)
Sommese, A., Wampler, C.: Numerical Solution of Polynomial Systems Arising in Engineering and Science. World Scientific, Singapore (2005)
Teitelbaum, J.: The computational complexity of the resolution of plane curve singularities. Math. Comput. 54(190), 797–837 (1990)
Wall, C.: Singular Points of Plane Curves. London Mathematical Society, London (2004)
Walsh, P.G.: A polynomial-time complexity bound for the computation of the singular part of an algebraic function. Math. Comput. 69, 1167–1182 (2000)
Zariski, O.: Studies in equisingularity i: equivalent singularities of plane algebroid curves. Am. J. Math. 87, 507–535 (1965)
Zariski, O.: Studies in equisingularity ii: equisingularity in codimension 1 (and characteristic zero). Am. J. Math. 87, 972–1006 (1965)
Zariski, O.: Studies in equisingularity iii: saturation of local rings and equisingularity. Am. J. Math. 90, 961–1023 (1968)
Zariski, O.: Le problème des modules pour les branches planes. Centre de Maths de l’X, Palaiseau (1973)
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Poteaux, A., Weimann, M. Computing the equisingularity type of a pseudo-irreducible polynomial. AAECC 31, 435–460 (2020). https://doi.org/10.1007/s00200-020-00451-x
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DOI: https://doi.org/10.1007/s00200-020-00451-x