Spiders and multiplicity sequences

Abhyankar, Shreeram; Luengo, Ignacio

doi:10.1090/S0002-9939-2013-12025-3

Spiders and multiplicity sequences
HTML articles powered by AMS MathViewer

by Shreeram S. Abhyankar and Ignacio Luengo PDF

Proc. Amer. Math. Soc. 141 (2013), 4071-4085 Request permission

Abstract:

The spider principle is used for establishing a formula for a finite quadratic sequence which determines the multiplicity sequences of all the sprouts which are founded upon the given finite quadratic sequence. This formula is basic for the theories of curvettes and dicriticals.

References

Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348. MR 82477, DOI 10.2307/2372519
Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, No. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416, DOI 10.1515/9781400881390
S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523, DOI 10.1007/978-3-662-03580-1
Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409–448. MR 401754, DOI 10.2307/2318338
Shreeram S. Abhyankar, Desingularization of plane curves, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 1–45. MR 713043
Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991, DOI 10.1090/surv/035
Shreeram S. Abhyankar, Polynomial expansion, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1583–1596. MR 1443142, DOI 10.1090/S0002-9939-98-04183-5
S. S. Abhyankar, Lectures on algebra. Vol. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2253320, DOI 10.1142/9789812773449
Shreeram S. Abhyankar, Inversion and invariance of characteristic terms: Part I, The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 93–168. MR 2744259, DOI 10.1007/978-1-4419-6263-8_{8}
Shreeram S. Abhyankar, Dicritical divisors and Jacobian problem, Indian J. Pure Appl. Math. 41 (2010), no. 1, 77–97. MR 2650101, DOI 10.1007/s13226-010-0017-x
Shreeram S. Abhyankar, Pillars and towers of quadratic transformations, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3067–3082. MR 2811263, DOI 10.1090/S0002-9939-2011-10731-7
Shreeram S. Abhyankar, More about dicriticals, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3083–3097. MR 2811264, DOI 10.1090/S0002-9939-2011-10732-9
Shreeram S. Abhyankar, Quadratic transforms inside their generic incarnations, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4111–4126. MR 2957201, DOI 10.1090/S0002-9939-2012-11583-7
S. S. Abhyankar, Dicritical divisors and Dedekind’s Gauss lemma, Proc. Amer. Math. Soc., to appear.
S. S. Abhyankar and E. Artal Bartolo, Curvettes and dicritical divisors, Proc. Amer. Math. Soc., to appear.
Shreeram S. Abhyankar and William J. Heinzer, Existence of dicritical divisors, Amer. J. Math. 134 (2012), no. 1, 171–192. MR 2876143, DOI 10.1353/ajm.2012.0002
Shreeram S. Abhyankar and William J. Heinzer, Existence of dicritical divisors revisited, Proc. Indian Acad. Sci. Math. Sci. 121 (2011), no. 3, 267–290. MR 2867982, DOI 10.1007/s12044-011-0035-6
Shreeram S. Abhyankar and Ignacio Luengo, Algebraic theory of dicritical divisors, Amer. J. Math. 133 (2011), no. 6, 1713–1732. MR 2863374, DOI 10.1353/ajm.2011.0045
E. Artal-Bartolo, Une démonstration géométrique du théorème d’Abhyankar-Moh, J. Reine Angew. Math. 464 (1995), 97–108 (French). MR 1340336, DOI 10.1515/crll.1995.464.97