Artal Bartolo, EnriqueLuengo Velasco, IgnacioMelle Hernández, Alejandro2023-06-182023-06-182015-11S.S. Abhyankar, Lectures on expansion techniques in algebraic geometry, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay, 1977, Notes by Balwant Singh. Dicritical divisors and Jacobian problem, Indian J. Pure Appl. Math. 41 (2010), no. 1, 77–97. More about dicriticals, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3083–3097. Pillars and towers of quadratic transformations, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3067–3082. Quadratic transforms inside their generic incarnations, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4111–4126. Dicriticals of pencils and Dedekind’s Gauss lemma, Rev. Mat. Complut. 26 (2013), no. 2, 735–752. Generic incarnations of quadratic transforms, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4103–4117. S.S. Abhyankar and E. Artal, Algebraic theory of curvettes and dicriticals, Proc. Amer. Math. Soc. 141 (2013),no. 12, 4087–4102. Analytic theory of curvettes and dicriticals, to appear in Rev. Mat. Complut., doi:10.1007/s13163-014-0148-4, 2014. S.S. Abhyankar and W.J. Heinzer, Existence of dicritical divisors revisited, Proc. Indian Acad. Sci. Math. Sci. 121 (2011), no. 3, 267–290. Existence of dicritical divisors, Amer. J. Math. 134 (2012), no. 1, 171–192. Rees valuations, Proc. Indian Acad. Sci. Math. Sci. 122 (2012), no. 4, 525–546. S.S. Abhyankar and I. Luengo, Algebraic theory of dicritical divisors, Amer. J. Math. 133 (2011), no. 6, 1713–1732. Spiders and multiplicity sequences, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4071–4085. S.S. Abhyankar and T.T. Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II, J. Reine Angew. Math. 260 (1973), 47–83; ibid. 261 (1973), 29–54. E. Artal, Une démonstration géométrique du théorème d’Abhyankar-Moh, J. Reine Angew. Math. 464 (1995),97–108. E. Artal, Pi. Cassou-Noguès, I. Luengo, and A. Melle-Hernández, On ν-quasi-ordinary power series: factorization, Newton trees and resultants, Topology of algebraic varieties and singularities, Contemp. Math., vol. 538, Amer.Math.Soc., Providence, RI, 2011, pp. 321–343. Pi. Cassou-Noguès, Newton trees at infinity of algebraic curves, Affine algebraic geometry, CRM Proc. Lecture Notes, vol. 54, Amer. Math. Soc., Providence, RI, 2011, pp. 1–19. A.H. Durfee, Five definitions of critical point at infinity, Singularities (Oberwolfach, 1996) (Basel), Progr. Math., no. 162, Birkhäuser, 1998, pp. 345–360. W. Engel, Ein Satz über ganze Cremona-Transformationen der Ebene, Math. Ann. 130 (1955), 11–19. R. Ephraim, Special polars and curves with one place at infinity, Singularities, Part 1 (Arcata, Calif., 1981), Proc.Sympos. Pure Math., vol. 40, Amer. Math.Soc.Providence, RI, 1983, pp. 353–359. J. Gwoździewicz, Ephraim’s pencils, Int. Math. Res. Not. IMRN (2013), no. 15, 3371–3385. Z. Jelonek and M. Tibar, Bifurcation locus and branches at infinity of a polynomial f : C2 → C, Preprint available at arXiv:1401.6544v1 [math.AG]. D.T. Lê and C. Weber, Équisingularité dans les pinceaux de germes de courbes planes et C0-suffisance, Enseign.Math. (2) 43 (1997), no. 3-4, 355–380. A. Melle-Hernández and C.T.C. Wall, Pencils of curves on smooth surfaces, Proc. London Math. Soc. (3) 83 (2001), no. 2, 257–278. T.T. Moh, On analytic irreducibility at ∞ of a pencil of curves, Proc. Amer. Math. Soc. 44 (1974), 22–24. On the Jacobian conjecture and the configurations of roots, J. Reine Angew. Math. 340 (1983), 140–212. W.A. Stein et al., Sage Mathematics Software (Version 5.10), The Sage Development Team, 2013,http://www.sagemath.org. H. Żołądek, An application of Newton-Puiseux charts to the Jacobian problem, Topology 47 (2008), no. 6, 431–469.0219-4988htpp://dx.doi.org/10.1142/S0219498815400095https://hdl.handle.net/20.500.14352/23022In this work we deal with dicritical divisors, curvettes and polynomials.These objects have been one of the main research interests of S.S. Abhyanka during his last years. In this work we provide some elementary proofs of some S.S.Abhyankar and I. Luengo results for dicriticals in the framework of forma power series. Based on these ideas we give anconstructive way to find the atypical fibres of a special pencil and give bounds for its number, which are sharper than the existing ones. Finally, we answer a question of J. Gwozdziewicz finding polynomials that reach his bound.engjournal articlehttp://www.worldscientific.com/doi/abs/10.1142/S0219498815400095http://arxiv.org/pdf/1408.0743.pdfopen access512.7Dicritical DivisorSpecial PencilGeometria algebraica1201.01 Geometría Algebraica