Publication: Approximate roots, toric resolutions and deformations of a plane branch
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2010-07
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Math Soc Japan
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We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial f is an element of C{x}[y], defining a plane branch (C, 0), in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non-equisingular) deformations of a plane branch (C, 0) supported on certain monomials in the approximate roots of f, which are essential in the study of Harnack smoothings of real plane branches by Risler and the author. Our results provide also a geometrical approach to Abhyankar's irreducibility criterion for power series in two variables and also a criterion to determine if a family of plane curves is equisingular to a plane branch.
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Correction of a proof in the paper “Approximate roots, toric resolutions and deformations of a plane branch” in: vol 65, pg 773, 2013
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N. A’Campo and M. Oka, Geometry of plane curves via Tschirnhausen resolution tower, Osaka J. Math., 33 (1996), 1003–1033.
S. S. Abhyankar, Lectures on expansion techniques in algebraic geometry, Notes by Balwant Singh, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 57, Tata Institute of Fundamental Research, Bombay, 1977.
S. S. Abhyankar, On the semigroup of a meromorphic curve, I, Proceedings of the International Symposium on Algebraic Geometry, Kyoto Univ., Kyoto, 1977, Kinokuniya Book Store, Tokyo, 1978, pp. 249–414.
S. S. Abhyankar, Irreducibility criterion for germs of analytic functions of two complex variables, Adv. Math., 74 (1989), 190–257.
S. S. Abhyankar and T. Moh, Newton-Puiseux Expansion and Generalized Tschirnhausen Transformation I–II, J. Reine Angew. Math., 260 (1973), 47–83; 261 (1973), 29–54.
S. S. Abhyankar and T. Moh, Embeddings of the Line in the Plane, J. Reine Angew. Math., 276 (1975), 148–166.
A. Assi and M. Barile, Effective construction of irreducible curve singularities, Int. J. Math. Comput. Sci., 1 (2006), 125–149.
A. Campillo, Algebroid curves in positive characteristic, Lecture Notes in Mathematics, 813, Springer, 1980.
E. Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, 276, Cambridge University Press, Cambridge, 2000.
V. Cossart and G. Moreno-Socías, Irreducibility criterion: a geometric point of view, Valuation theory and its applications, II, Saskatoon, SK, 1999, Fields Inst. Commun., 33, Amer. Math. Soc., Providence, RI, 2003, pp. 27–42.
V. Cossart and G. Moreno-Socías, Racines approchées, suites génératrices, suffisance des jets, Ann. Fac. Sci. Toulouse Math., 14 (2005), 353–394.
E. R. García Barroso, Sur les courbes polaires d’une courbe plane réduite, Proc.London Math. Soc. (3), 81 (2000), 1–28.
E. R. García Barroso and J. Gwoździewicz, Characterization of jacobian Newton polygons of plane branches and new criteria of irreductibility, Ann. Inst. Fourier (Grenoble), 60 (2010), 683–709.
R. Goldin and B. Teissier, Resolving singularities of plane analytic branches with one toric morphism, Resolution of Singularities, A research textbook in tribute to Oscar Zariski, (eds. H. Hauser, J. Lipman, F. Oort and A. Quiros), Progr. Math.,181, Birkhäuser, Basel, 2000, pp. 315–340.
P. D. González Pérez, Toric embedded resolutions of quasi-ordinary hypersurface singularities, Ann. Inst. Fourier (Grenoble), 53 (2003), 1819–1881.
P. D. González Pérez and J.-J. Risler, Multi-Harnack smoothings of real plane branches, Ann. Sci. École Norm. Sup. (4), 43 (2010), 143–183.
J. Gwoździewicz and A. Ploski, On the approximate roots of polynomials, Ann. Polon. Mathe., 60 (1995), 199–210.
J. Gwoździewicz, Kouchnirenko type formulas for local invariants of plane analytic curves, arXiv:0707.3404v1 [math.AG].
D. T. Lê and M. Oka, On resolution complexity of plane curves, Kodai Math. J., 18 (1995), 1–36.
M. Merle, Invariants polaires des courbes planes, Invent. Math., 41 (1977), 103–111.
M. Oka, Geometry of plane curves via toroidal resolution, Algebraic Geometry and Singularities, (eds. A. Campillo López and L. Narváez Macarro), Progr. Math., 134, Birkhäuser, Basel, 1996, pp. 95–121.
M. Oka, Non-degenerate complete intersection singularity, Actualités Mathématiques, Hermann, Paris, 1997.
H. Pinkham, Courbes planes ayant une seule place a l’infini, Séminaire sur les Singularités des surfaces, Centre de Mathématiques de l’École Polytechnique, Année, 1977–1978.
P. Popescu-Pampu, Approximate roots, Valuation theory and its applications, II (Saskatoon, SK, 1999), Fields Inst. Commun., 33, Amer. Math. Soc., Providence, RI, 2003, pp. 285–321.
J.-J. Risler, Sur l’idéal jacobien d’une courbe plane, Bulletin de la S. M. F., 99 (1971), 305–311.
B. Teissier, Variétés polaires, I. Invariants polaires des singularités d’hypersurfaces, Invent. Math., 40 (1977), 267–292.
B. Teissier, The monomial curve and its deformations, Appendix in [Z2].
B. Teissier, Complex curve singularities: a biased introduction, Singularities in geometry and topology, World Sci. Publ., Hackensack, NJ, 2007, pp. 825–887.
C. T. C.Wall, Singular points of plane curves, London Mathematical Society Student Texts, 63, Cambridge University Press, Cambridge, 2004.
O. Zariski, Studies in equisingularity III, Saturation of local rings and equisingularity, Amer. J. Math., 90 (1968), 961–1023.
O. Zariski, Le problème des modules pour les branches planes, Hermann, Paris, 1986.