Resolution of singularities and modular Galois theory
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Abstract:
I shall sketch a brief history of the desingularization problem from Riemann thru Zariski to Hironaka, including the part I played in it and the work on Galois theory which this led me to, and how that caused me to search out many group theory gurus. I shall also formulate several conjectures and suggest numerous thesis problems.References
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Shreeram Abhyankar, Coverings of algebraic curves, Amer. J. Math. 79 (1957), 825–856. MR 94354, DOI 10.2307/2372438
- H. S. Vandiver, Certain congruences involving the Bernoulli numbers, Duke Math. J. 5 (1939), 548–551. MR 21, DOI 10.1215/S0012-7094-39-00546-6
- Shreeram S. Abhyankar, Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 111–152. MR 0200272
- Shreeram Shankar Abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure and Applied Mathematics, Vol. 24, Academic Press, New York-London, 1966. MR 0217069
- Shreeram Shankar Abhyankar, An algorithm on polynomials in one indeterminate with coefficients in a two dimensional regular local domain, Ann. Mat. Pura Appl. (4) 71 (1966), 25–59. MR 207699, DOI 10.1007/BF02413732
- Shreeram Shankar Abhyankar, Resolution of singularities of algebraic surfaces, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 1–11. MR 0257080
- 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, Weighted expansions for canonical desingularization, Lecture Notes in Mathematics, vol. 910, Springer-Verlag, Berlin-New York, 1982. With a foreword by U. Orbanz. MR 653634, DOI 10.1007/BFb0093060
- Shreeram S. Abhyankar, Determinantal loci and enumerative combinatorics of Young tableaux, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 1–26. MR 977749, DOI 10.1016/B978-0-12-348031-6.50007-3
- Shreeram S. Abhyankar, Enumerative combinatorics of Young tableaux, Monographs and Textbooks in Pure and Applied Mathematics, vol. 115, Marcel Dekker, Inc., New York, 1988. MR 926272
- 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 [A16]A16 S. S. Abhyankar, Invariant theory and enumerative combinatorics of Young tableaux, Geometric Invariance in Computer Vision, Edited by J. L. Mundy and A. Zisserman, MIT Press (1992), 45-76.
- Shreeram S. Abhyankar, Galois theory on the line in nonzero characteristic, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 68–133. MR 1118002, DOI 10.1090/S0273-0979-1992-00270-7
- Shreeram S. Abhyankar, Mathieu group coverings in characteristic two, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 3, 267–271 (English, with English and French summaries). MR 1205196, DOI 10.2307/2161167
- Shreeram S. Abhyankar, Alternating group coverings of the affine line for characteristic greater than two, Math. Ann. 296 (1993), no. 1, 63–68. MR 1213371, DOI 10.1007/BF01445094
- Shreeram S. Abhyankar, Square-root parametrization of plane curves, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 19–84. MR 1272021, DOI 10.1007/978-1-4612-2628-4_{2}
- Shreeram S. Abhyankar, Nice equations for nice groups, Israel J. Math. 88 (1994), no. 1-3, 1–23. MR 1303488, DOI 10.1007/BF02937504
- Shreeram S. Abhyankar, Mathieu group coverings and linear group coverings, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 293–319. MR 1352279, DOI 10.1090/conm/186/02188
- Shreeram S. Abhyankar, Fundamental group of the affine line in positive characteristic, Geometry and analysis (Bombay, 1992) Tata Inst. Fund. Res., Bombay, 1995, pp. 1–26. MR 1351500
- Shreeram S. Abhyankar, Again nice equations for nice groups, Proc. Amer. Math. Soc. 124 (1996), no. 10, 2967–2976. MR 1343675, DOI 10.1090/S0002-9939-96-03471-5
- Shreeram S. Abhyankar, More nice equations for nice groups, Proc. Amer. Math. Soc. 124 (1996), no. 10, 2977–2991. MR 1343676, DOI 10.1090/S0002-9939-96-03472-7
- Shreeram S. Abhyankar, Further nice equations for nice groups, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1555–1577. MR 1348146, DOI 10.1090/S0002-9947-96-01584-X
- Shreeram S. Abhyankar, Factorizations over finite fields, Finite fields and applications (Glasgow, 1995) London Math. Soc. Lecture Note Ser., vol. 233, Cambridge Univ. Press, Cambridge, 1996, pp. 1–21. MR 1433135, DOI 10.1017/CBO9780511525988.003
- Shreeram S. Abhyankar, Local fundamental groups of algebraic varieties, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1635–1641. MR 1403110, DOI 10.1090/S0002-9939-97-03938-5
- Shreeram S. Abhyankar, Projective polynomials, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1643–1650. MR 1403111, DOI 10.1090/S0002-9939-97-03939-7
- Shreeram S. Abhyankar, Hilbert’s thirteenth problem, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995) Sémin. Congr., vol. 2, Soc. Math. France, Paris, 1997, pp. 1–11 (English, with English and French summaries). MR 1601178
- 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, Polynomial expansion, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1583–1596. MR 1443142, DOI 10.1090/S0002-9939-98-04183-5
- Shreeram S. Abhyankar, Semilinear transformations, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2511–2525. MR 1676323, DOI 10.1090/S0002-9939-99-05400-3
- Shreeram S. Abhyankar, Galois theory of semilinear transformations, Aspects of Galois theory (Gainesville, FL, 1996) London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 1–37. MR 1708600 [A35]A35 S. S. Abhyankar, Galois embeddings for linear groups, Transactions of The American Mathematical Society 352 (2000), 3881-3912. [A36]A36 S. S. Abhyankar, Two step descent in modular Galois theory, therorems of Burnside and Caley, and Hilbert’s thirteenth problem, To Appear. [A37]A37 S. S. Abhyankar, Desingularizaton and modular Galois theory, To Appear.
- Shreeram S. Abhyankar, Stephen D. Cohen, and Michael E. Zieve, Bivariate factorizations connecting Dickson polynomials and Galois theory, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2871–2887. MR 1491853, DOI 10.1090/S0002-9947-00-02271-6
- Shreeram S. Abhyankar and Sudhir R. Ghorpade, Young tableaux and linear independence of standard monomials in multiminors of a multimatrix, Discrete Math. 96 (1991), no. 1, 1–32. MR 1139437, DOI 10.1016/0012-365X(91)90467-G
- Shreeram S. Abhyankar and Sanjeevani B. Joshi, Generalized codeletion and standard multitableaux, Group actions and invariant theory (Montreal, PQ, 1988) CMS Conf. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 1989, pp. 1–24. MR 1021271
- Shreeram S. Abhyankar and Sanjeevani B. Joshi, Generalized roinsertive correspondence between multitableaux and multimonomials, Discrete Math. 90 (1991), no. 2, 111–135. MR 1115498, DOI 10.1016/0012-365X(91)90350-B
- Shreeram S. Abhyankar and Sanjeevani B. Joshi, Generalized rodeletive correspondence between multitableux and multimonomials, Discrete Math. 93 (1991), no. 1, 1–17. MR 1141259, DOI 10.1016/0012-365X(91)90213-L
- Shreeram S. Abhyankar and Sanjeevani B. Joshi, Generalized coinsertion and standard multitableaux, J. Statist. Plann. Inference 34 (1993), no. 1, 5–18. MR 1209985, DOI 10.1016/0378-3758(93)90029-6 [AKe]AKe S. S. Abhyankar and P. H.Keskar, Descent principle in modular Galois theory, To Appear.
- Shreeram Abhyankar and Devadatta M. Kulkarni, On Hilbertian ideals, Linear Algebra Appl. 116 (1989), 53–79. MR 989717, DOI 10.1016/0024-3795(89)90398-4
- Shreeram S. Abhyankar and Devadatta M. Kulkarni, Bijection between indexed monomials and standard bitableaux, Discrete Math. 79 (1989/90), no. 1, 1–48. MR 1032632, DOI 10.1016/0012-365X(90)90053-K
- Shreeram S. Abhyankar and Devadatta M. Kulkarni, Coinsertion and standard bitableaux, Discrete Math. 85 (1990), no. 2, 115–166. MR 1080622, DOI 10.1016/0012-365X(90)90018-D
- Shreeram S. Abhyankar and Paul A. Loomis, Once more nice equations for nice groups, Proc. Amer. Math. Soc. 126 (1998), no. 7, 1885–1896. MR 1459101, DOI 10.1090/S0002-9939-98-04421-9 [AL2]AL2 S. S. Abhyankar and P. A. Loomis, Twice more nice equations for nice groups, Contemporary Mathematics 245 (1999), 63-76.
- Shreeram S. Abhyankar, Jun Ou, and Avinash Sathaye, Alternating group coverings of the affine line for characteristic two, Discrete Math. 133 (1994), no. 1-3, 25–46. MR 1298962, DOI 10.1016/0012-365X(94)90014-0
- Shreeram S. Abhyankar, Wolfgang K. Seiler, and Herbert Popp, Mathieu-group coverings of the affine line, Duke Math. J. 68 (1992), no. 2, 301–311. MR 1191563, DOI 10.1215/S0012-7094-92-06813-X
- Shreeram S. Abhyankar and Ganapathy S. Sundaram, Galois theory of Moore-Carlitz-Drinfeld modules, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 4, 349–353 (English, with English and French summaries). MR 1467085, DOI 10.1016/S0764-4442(97)85615-7 [AS2]AS2 S. S. Abhyankar and G. S. Sundaram, Galois groups of generalized iterates of generic vectorial polynomials, To Appear.
- Shreeram S. Abhyankar and Ikkwon Yie, Some more Mathieu group coverings in characteristic two, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1007–1014. MR 1239794, DOI 10.1090/S0002-9939-1994-1239794-1
- Shreeram S. Abhyankar and Ikkwon Yie, Small Mathieu group coverings in characteristic two, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1319–1329. MR 1246511, DOI 10.1090/S0002-9939-1995-1246511-9 [Alb]Alb G. Albanese, Transformazione birazionale di una superficie algebriche in un’altra priva di punti multipli, Rendiconti della Circolo Matematica de Palermo 48 (1924), 321-332.
- M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- Francis Buekenhout and Ernest Shult, On the foundations of polar geometry, Geometriae Dedicata 3 (1974), 155–170. MR 350599, DOI 10.1007/BF00183207 [Bur]Bur W. Burnside, Theory of groups of finite order, Cambridge University Press, First Edition 1897, Second Edition 1911. [BPa]BPa W. S. Burnside and A. W. Panton, Theory of Equations I–II, Dublin, Hodges, Figgis and Co., London, 1904.
- Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), no. 1, 1–22. MR 599634, DOI 10.1112/blms/13.1.1
- P. J. Cameron and W. M. Kantor, $2$-transitive and antiflag transitive collineation groups of finite projective spaces, J. Algebra 60 (1979), no. 2, 384–422. MR 549937, DOI 10.1016/0021-8693(79)90090-5 [Ca1]Ca1 L. Carlitz, A class of polynomials, Transactions of the American Mathematical Society 43 (1938), 167-182.
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3 [Cay]Cay A. Cayley, On the intersection of curves, Mathematische Annalen 30 (1887), 85-90.
- Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088, DOI 10.1007/BF03046993
- Stephen D. Cohen and Rex W. Matthews, A class of exceptional polynomials, Trans. Amer. Math. Soc. 345 (1994), no. 2, 897–909. MR 1272675, DOI 10.1090/S0002-9947-1994-1272675-0
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938 [Cre]Cre L. Cremona, Elementi di Geometria Projecttiva, Rome/Turin: G. B. Paravia and Company, 1873.
- Steven Dale Cutkosky, Local factorization of birational maps, Adv. Math. 132 (1997), no. 2, 167–315. MR 1491444, DOI 10.1006/aima.1997.1675 [Cu2]Cu2 S. D. Cutkosky, Local monomialization and factorization of morphisms, Asterisque, No. 260 (1999). [DWe]DWe R. Dedekind and H. Weber, Theorie der algebraischen functionen einer veränderlichen, Crelle Journal 92 (1882), 181-290. [Dic]Dic L. E. Dickson, Linear Groups, Teubner, 1901.
- V. G. Drinfel′d, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian). MR 0384707
- Walter Feit, On a class of doubly transitive permutation groups, Illinois J. Math. 4 (1960), 170–186. MR 113953 [For]For A. R. Forsyth, Theory of Functions of a Complex Variable, Cambridge University Press, London, 1893.
- Michael Fried, On Hilbert’s irreducibility theorem, J. Number Theory 6 (1974), 211–231. MR 349624, DOI 10.1016/0022-314X(74)90015-8
- Michael D. Fried, Robert Guralnick, and Jan Saxl, Schur covers and Carlitz’s conjecture, Israel J. Math. 82 (1993), no. 1-3, 157–225. MR 1239049, DOI 10.1007/BF02808112
- David Fu, Local weak simultaneous resolution for high rational ranks, J. Algebra 194 (1997), no. 2, 614–630. MR 1467169, DOI 10.1006/jabr.1996.7014
- E.-U. Gekeler, Moduli for Drinfel′d modules, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 153–170. MR 1196518
- Daniel Gorenstein, Classifying the finite simple groups, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 1, 1–98. MR 818060, DOI 10.1090/S0273-0979-1986-15392-9
- David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131, DOI 10.1007/978-3-642-61480-4
- Hermann Weyl, Invariants, Duke Math. J. 5 (1939), 489–502. MR 30 [Gur]Gur R.M. Guralnick, Monodromy groups of covers of small genus in positive characteristic, To Appear.
- Robert M. Guralnick and Jan Saxl, Monodromy groups of polynomials, Groups of Lie type and their geometries (Como, 1993) London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 125–150. MR 1320519, DOI 10.1017/CBO9780511565823.012 [GSt]GSt R.M. Guralnick and K. F. Stevenson, Prescribing ramification, To Appear.
- David Harbater, Abhyankar’s conjecture on Galois groups over curves, Invent. Math. 117 (1994), no. 1, 1–25. MR 1269423, DOI 10.1007/BF01232232
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- D. R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77–91. MR 330106, DOI 10.1090/S0002-9947-1974-0330106-6
- Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geometriae Dedicata 2 (1974), 425–460. MR 335659, DOI 10.1007/BF00147570
- Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. Algebra 93 (1985), no. 1, 151–164. MR 780488, DOI 10.1016/0021-8693(85)90179-6 [Hi1]Hi1 D. Hilbert, Mathematische Probleme, Archiv für Mathematik und Physik 1 (1901), 44-63 and 213-237. [Hi2]Hi2 D. Hilbert, Über die Gleichung neunten Grades, Mathematische Annalen 97 (1927), 243-250.
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
- Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7 [Jor]Jor C. Jordan, Traité des Substitutions et des Équatione Algébriques, Gauthier-Villars, 1870. [Jun]Jun H. W. E. Jung, Darstellung der Funktionen eines algebraischen Körpers zweier Veränderlichen $x,y$ in der Umgebung einer Stelle $x = a, y = b$, Crelle Journal 133 (1908), 289-314.
- William M. Kantor, Rank $3$ characterizations of classical geometries, J. Algebra 36 (1975), no. 2, 309–313. MR 387386, DOI 10.1016/0021-8693(75)90106-4
- William M. Kantor, Homogeneous designs and geometric lattices, J. Combin. Theory Ser. A 38 (1985), no. 1, 66–74. MR 773556, DOI 10.1016/0097-3165(85)90022-6
- Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235 [Kle]Kle F. Klein, Entwicklung der Mathematik im neunzehnten Jahrhundert, Berlin, 1926. [Kr1]Kr1 W. Krull, Allgemeine Bewertungstheorie, Crelle Journal 167 (1932), 160-196. [Kr2]Kr2 W. Krull, Dimensionstheorie in Stellenringe, Crelle Journal 179 (1938), 204-226.
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335 [Len]Len H. W. Lenstra, On the degrees of permutation polynomials, Abstracts of the Third International Conference on Finite Fields and Applications (1995), 40.
- H. W. Lenstra Jr. and M. Zieve, A family of exceptional polynomials in characteristic three, Finite fields and applications (Glasgow, 1995) London Math. Soc. Lecture Note Ser., vol. 233, Cambridge Univ. Press, Cambridge, 1996, pp. 209–218. MR 1433150, DOI 10.1017/CBO9780511525988.018
- Martin W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), no. 3, 477–516. MR 879395, DOI 10.1112/plms/s3-54.3.477
- Martin W. Liebeck, Characterization of classical groups by orbit sizes on the natural module, Proc. Amer. Math. Soc. 124 (1996), no. 10, 2961–2966. MR 1343709, DOI 10.1090/S0002-9939-96-03505-8
- Martin W. Liebeck and Gary M. Seitz, On the subgroup structure of classical groups, Invent. Math. 134 (1998), no. 2, 427–453. MR 1650328, DOI 10.1007/s002220050270
- Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1711577, DOI 10.1007/978-3-662-12123-8 [Mar]Mar B. Marggraff, Über primitive Gruppen mit transitiven Untergruppen geringeren Grades, Giessen Dissertation (1892). [Mat]Mat E. Mathieu, Mémoire sur l’étude des fonctions de plusieurs quantités sur la manière de les former, et sur les substitutions qui les laissent invariables, J. Math Pures Appl. 6 (1861), 241-323.
- M. R. Modak, Combinatorial meaning of the coefficients of a Hilbert polynomial, Proc. Indian Acad. Sci. Math. Sci. 102 (1992), no. 2, 93–123. MR 1195735, DOI 10.1007/BF02836782 [Moo]Moo E. H. Moore, A two-fold generalization of Fermat’s theorem, Bulletin of the American Mathematical Society 2 (1896), 189-199.
- Peter Müller, New examples of exceptional polynomials, Finite fields: theory, applications, and algorithms (Las Vegas, NV, 1993) Contemp. Math., vol. 168, Amer. Math. Soc., Providence, RI, 1994, pp. 245–249. MR 1291433, DOI 10.1090/conm/168/01704
- S. B. Mulay, Determinantal loci and the flag variety, Adv. Math. 74 (1989), no. 1, 1–30. MR 991408, DOI 10.1016/0001-8708(89)90002-9 [Neu]Neu C. Neumann, Riemann’s Theorie der Abel’schen Integrale, Teubner, 1888. [New]New I. Newton, Geometria Analitica, 1660. [NoE]NoE E. Noether, Eliminationstheorie und allgemeine Idealtheorie, Mathematische Annalen 90 (1923), 229-261. [NoM]NoM M. Noether, Über einen Satz aus der Theorie der algebraischen Funktionen, Mathematische Annalen 6 (1873), 351-359. [Ost]Ost A. Ostrowski, Über einige Lösungen der Funktionalgleichung $\phi (x) \phi (y) =\phi (xy)$, Acta Mathematica 41 (1918), 271-284. [Pri]Pri R. J. Pries, Formal patching and deformation of wildly ramified covers of curves, University of Pennsylvania Thesis (2000).
- M. Raynaud, Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar, Invent. Math. 116 (1994), no. 1-3, 425–462 (French). MR 1253200, DOI 10.1007/BF01231568 [Rie]Rie B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inauguraldissertation, Göttingen (1851), 1-48. [Sal]Sal G. Salmon, Higher Plane Curves, Dublin, 1852.
- David L. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294–320. MR 330154, DOI 10.2307/2373787 [Sev]Sev F. Severi, Vorlesungen über algebraische Geometrie, Teubner, 1921.
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086 [Se2]Se2 J.-P. Serre, Résumé de cours et travaux, Annuaire du Collège de France 85-86 (1985).
- J.-P. Serre, A letter as an appendix to the square-root parameterization paper of Abhyankar, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 85–88. MR 1272022 [Spe]Spe A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, Berlin, 1937. [Sta]Sta H. Stahl, Theorie der Abel’schen Funktionen, Teubner, 1896.
- Michio Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. MR 136646, DOI 10.2307/1970423 [Syl]Syl J. J. Sylvester, On a general method of determining by mere inspection the derivations from two equations of any degree, Philosophical Magazine 16 (1840).
- Dinesh S. Thakur, Drinfel′d modules and arithmetic in the function fields, Internat. Math. Res. Notices 9 (1992), 185–197. MR 1185833, DOI 10.1155/S1073792892000217
- John G. Thompson, Some finite groups which appear as $\textrm {Gal}\,L/K$, where $K\subseteq \textbf {Q}(\mu _{n})$, J. Algebra 89 (1984), no. 2, 437–499. MR 751155, DOI 10.1016/0021-8693(84)90228-X
- John G. Thompson, Rigidity, $\textrm {GL}(n,q)$, and the braid group, Bull. Soc. Math. Belg. Sér. A 42 (1990), no. 3, 723–733. Algebra, groups and geometry. MR 1316220
- Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
- Shrinivas G. Udpikar, On Hilbert polynomial of certain determinantal ideals, Internat. J. Math. Math. Sci. 14 (1991), no. 1, 155–162. MR 1087410, DOI 10.1155/S0161171291000157 [VYo]VYo O. Veblen and J. T. Young, Projective Geometry I–II, Ginn and Company, 1910-1918.
- F. D. Veldkamp, Polar geometry. I, II, III, IV, V, Nederl. Akad. Wetensch. Proc. Ser. A 62; 63 = Indag. Math. 21 (1959), 512-551 22 (1959), 207–212. MR 0125472
- Helmut Völklein, $\textrm {GL}_n(q)$ as Galois group over the rationals, Math. Ann. 293 (1992), no. 1, 163–176. MR 1162680, DOI 10.1007/BF01444710
- Helmut Völklein, Braid group action via $\textrm {GL}_n(q)$ and $\textrm {U}_n(q)$, and Galois realizations, Israel J. Math. 82 (1993), no. 1-3, 405–427. MR 1239059, DOI 10.1007/BF02808122
- Helmut Völklein, Groups as Galois groups, Cambridge Studies in Advanced Mathematics, vol. 53, Cambridge University Press, Cambridge, 1996. An introduction. MR 1405612, DOI 10.1017/CBO9780511471117 [Wan]Wan D. Wan, A generalization of the Carlitz conjecture, Finite Fields, Coding Theory and Advances in Communications and Computing, Lecture Notes in Pure and Applied Mathematics, Dekker 141 (1993), 431-432.
- Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775 [Wey]Wey H. Weyl, Die Idee der Riemannschen Fläche, Teubner, 1923. [Za1]Za1 O. Zariski, Some results in the arithmetical theory of algebraic varieties, American Journal of Mathematics 61 (1939), 224-294. [Za2]Za2 O. Zariski, The reduction of singularities of algebraic surfaces, Annals of Mathematics 40 (1939), 639-689.
- Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335 [Z01]Z01 H. J. Zassenhaus, Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen, Abh. Math. Sem, Univ of Hamburg 11 (1936), 17-44.
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Additional Information
- Shreeram S. Abhyankar
- Affiliation: Mathematics Department, Purdue University, West Lafayette, IN 47907
- Email: ram@cs.purdue.edu
- Received by editor(s): December 7, 1999
- Published electronically: December 27, 2000
- Additional Notes: Abhyankar’s work was partly supported by NSF Grant DMS 97-32592 and NSA grant MDA 904-97-1-0010.
- © Copyright 2000 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 38 (2001), 131-169
- MSC (2000): Primary 12F10, 14H30, 20D06, 20E22
- DOI: https://doi.org/10.1090/S0273-0979-00-00892-2
- MathSciNet review: 1816069