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Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification

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Abstract

Davenport’s Problem asks: What can we expect of two polynomials, over Z, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, Lewis and Schinzel.

By bounding the degrees, but expanding the maps and variables in Davenport’s Problem, Galois stratification enhanced the separated variable theme, solving an Ax and Kochen problem from their Artin Conjecture work. Denef and Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement.

By restricting the variables, but leaving the degrees unbounded, we found the striking distinction between Davenport’s problem over Q, solved by applying the Branch Cycle Lemma, and its generalization over any number field, solved by using the simple group classification. This encouraged Thompson to formulate the genus 0 problem on rational function monodromy groups. Guralnick and Thompson led its solution in stages. We look at two developments since the solution of Davenport’s problem.

  • Stemming from MacCluer’s 1967 thesis, identifying a general class of problems, including Davenport’s, as monodromy precise.

  • R(iemann)E(xistence)T(heorem)’s role as a converse to problems generalizing Davenport’s, and Schinzel’s (on reducibility).

We use these to consider: Going beyond the simple group classification to handle imprimitive groups, and what is the role of covers and correspondences in going from algebraic equations to zeta functions with Chow motive coefficients.

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References

  1. Abhyankar S. Coverings of algebraic curves. Amer J Math, 1957, 79: 825–856

    Article  MATH  MathSciNet  Google Scholar 

  2. Abhyankar S. Projective polynomials. Proc Amer Math Soc, 1997, 125: 1643–1650

    Article  MATH  MathSciNet  Google Scholar 

  3. Agricola I. Old and new on the exceptional group G 2. Notice Amer Math Soc, 2008, 55: 922–929

    MATH  MathSciNet  Google Scholar 

  4. Ahlfors L. Complex Analysis: an introduction to the theory of analytic functions of one complex variable, 3rd ed. In: Series in Pure and Applied Math. New York: The McGraw-Hill Companies, 1979

    Google Scholar 

  5. Aitken W. On value sets of polynomials over a finite field. Finite Fields Appl, 1998, 4: 441–449

    Article  MATH  MathSciNet  Google Scholar 

  6. Artin E. Über die Zetafuncktionen gewisser algebraischer Zahlkörper. Math Ann, 1923, 89: 147–156

    Article  MATH  MathSciNet  Google Scholar 

  7. Artin E. Geometric Algebra. New York: Interscience, 1957

    MATH  Google Scholar 

  8. Aschbacher M, Scott L. Maximal subgroups of finite groups. J Algebra, 1985, 92: 44–80

    Article  MATH  MathSciNet  Google Scholar 

  9. Avanzi R M, Zannier U M. Genus one curves defined by separated variable polynomials and a polynomial Pell equation. Acta Arith, 2001, 99: 227–256

    Article  MATH  MathSciNet  Google Scholar 

  10. Avanzi R M, Zannier U M. The Equation f(X) = f(Y) in Rational Functions X = X(t), Y = Y (t). Comp Math Kluwer Acad, 2003, 139: 263–295

    MATH  MathSciNet  Google Scholar 

  11. Ax J. The elementary theory of finite fields. Ann of Math, 1968, 88: 239–271

    Article  MATH  MathSciNet  Google Scholar 

  12. Ax J. A mathematical approach to some problems in number theory. Proceedings of Symposia in Pure Mathmatics, vol. 20. Providence, RI: Amer Math Soc, 1971, 161–190

    Google Scholar 

  13. Ax J, Kochen S. Diophantine problems over local fields III (culminating paper of the series). Ann of Math, 1966, 83: 437–456

    Article  MATH  MathSciNet  Google Scholar 

  14. Bailey P, Fried M D. Hurwitz monodromy, spin separation and higher levels of a Modular Tower. In: Proceedings of Symposia in Pure Mathmatics, vol.70. Providence, R I: Amer Math Soc, 2002

    Google Scholar 

  15. Beardon A, Ng T. Parameterizations of algebraic curves. Ann Acad Sci Fenn Math, 2006, 31: 541–554

    MATH  MathSciNet  Google Scholar 

  16. Beke T. Zeta functions of equivalence relations over finite fields. Finite Fields Appl, 2011, 17: 68–80

    Article  MATH  MathSciNet  Google Scholar 

  17. Beukers F, Shorey T N, Tijdeman R. Irreducibility of polynomials and arithmetic progressions with equal products of terms. No Th in Prog. Berlin-New York: Walter de Gruyter, 1999

    Google Scholar 

  18. Biggers R, Fried M. Moduli spaces of covers and the Hurwitz monodromy group. Crelles J, 1982, 335: 87–121

    Article  MATH  MathSciNet  Google Scholar 

  19. Biggers R, Fried M. Irreducibility of moduli spaces of cyclic unramified covers of genus g curves. Trans Amer Math Soc, 1986, 295: 59–70

    MATH  MathSciNet  Google Scholar 

  20. Bilu Y F. Quadratic factors of f(x) − g(y). Acta Arith, 1999, 90: 341–355

    MATH  MathSciNet  Google Scholar 

  21. Bilu Y F, Tichy R F. The diophantine equation f(x) − g(y). Acta Arith, 2000, 95: 261–288

    MATH  MathSciNet  Google Scholar 

  22. Bluher A. Explicit formulas for strong Davenport pairs. Act Arith, 2004, 112.4: 397–403

    Article  MathSciNet  Google Scholar 

  23. Bombieri E. On exponential sum in finite fields II. Invent Math, 1978, 47: 20–39

    Article  MathSciNet  Google Scholar 

  24. Borevich Z I, Shafarevich I R. Number Theory. New York: Academic Press, 1966

    Google Scholar 

  25. Carmichael R. Introduction to the Theory of Groups of Finite Order. New York: Dover Publications, 1956

    MATH  Google Scholar 

  26. Cassels J, Fröhlich A. Algebraic Number Theory. Washington D C: Thompson Book Co., 1967

    MATH  Google Scholar 

  27. Fuchs C, Zannier U. Composite rational functions expressible with few terms. Preprint, 2010

  28. Cohen S D. Exceptional polynomials and the reducibility of substitution polynomials. Enseign Math, 1990, 36: 309–318

    MathSciNet  Google Scholar 

  29. Cohen S D, Fried M D. Lenstra’s proof of the Carlitz-Wan conjecture on exceptional polynomials: an elementary version. Finite Fields Appl, 1995, 1: 372–375

    Article  MATH  MathSciNet  Google Scholar 

  30. Conway J B. Functions of a complex variable, 2nd ed. New York: Springer-Verlag Grad text, 1978

    Book  Google Scholar 

  31. Cohen S D, Matthews R W. A class of exceptional polynomials. Trans Amer Math Soc, 1994, 345: 897–909

    Article  MATH  MathSciNet  Google Scholar 

  32. Paths that are classical generators of the punctured sphere: http://math.uci.edu/?mfried/deflist-cov/classicalgens.pdf. The genus 0 problem for rational functions: http://math.uci.edu/?mfried/deflist-cov/Genus0-Prob.html

  33. Cox D. What is the Role of Algebra in Applied Mathematics? Notices Amer Math Soc, 2005: 1193–1198

  34. Couveignes J M, Cassou-Nogus P. Factorisations explicites de g(y) − h(z). Acta Arith, 1999, 87: 291–317

    MATH  MathSciNet  Google Scholar 

  35. Curtis C W, Kantor W M, Seitz G M. The 2-transitive permutation representations of the finite Chevalley groups. Trans Amer Math Soc, 1976, 218: 1–59

    MATH  MathSciNet  Google Scholar 

  36. Davenport H, Lewis D J, Schinzel A. Equations of Form f(x) = g(y). Quart J Math Oxford, 1961, 12: 304–312

    Article  MATH  MathSciNet  Google Scholar 

  37. Davenport H, Lewis D J. Notes on Congruences (I). Quart J Math Oxford, 1963, 14: 51–60

    Article  MATH  MathSciNet  Google Scholar 

  38. Dèbes P. Arithmétique et espaces de modules de revêvetements. In: Number Theory in Progress. Berlin-New York: Walter de Gruyter, 1999

    Google Scholar 

  39. Dèbes P. Arithmétique des revêtements de la droite. http://math.univ-lille1.fr/?de/pub.html

  40. Dèbes P, Fried M. Rigidity and real residue class fields. Acta Arith, 1990, 56: 13–45

    Google Scholar 

  41. Dèbes P, Fried M D. Arithmetic variation of fibers in families: Hurwitz monodromy criteria for rational points on all members of the family. Crelles J, 1990, 409: 106–137

    MATH  Google Scholar 

  42. Dèbes P, Fried M D. Nonrigid situations in constructive Galois theory. Pacific J Math, 1994, 163: 81–122

    MATH  MathSciNet  Google Scholar 

  43. Dèbes P, Fried M D. Integral Specialization of families of rational functions. Pacific J Math, 1999, 190: 75–103

    Article  Google Scholar 

  44. Deligne P, Mumford D. The irreducibility of the space of curves of given genus. Publ Math IHES, 1969, 36: 75–100

    MATH  MathSciNet  Google Scholar 

  45. Deligne P. La conjecture de Weil I. Publ Math IHES, 1974, 43: 273–307

    MathSciNet  Google Scholar 

  46. Deligne P. La conjecture de Weil: II. Publ Math IHES, 1980, 52: 137–252

    MATH  MathSciNet  Google Scholar 

  47. Deligne P. Le Groupe fondamental de la Droite Projective Moins Trois Points. In: Galois Groups over Q. New York: Springer-Verlag, 1989

    Google Scholar 

  48. Denef J. The rationality of the Poincaré series associated to the p-adic points on a variety. Invent Math, 1984, 77: 1–23

    Article  MATH  MathSciNet  Google Scholar 

  49. Denef J, Loeser F. Definable sets, motives and p-adic integrals., 2001, 14: 429–4

    MATH  MathSciNet  Google Scholar 

  50. Dwork B. On the zeta function of a hypersurface III. Ann of Math, 1966, 83: 457–519

    Article  MathSciNet  Google Scholar 

  51. Evertse J H. Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces. http://front.math.ucdavis.edu/ANT

  52. Feit W. Automorphisms of symmetric balanced incomplete block designs. Math Zeit, 1970, 118: 40–49

    Article  MATH  MathSciNet  Google Scholar 

  53. Feit W. Automorphisms of symmetric balanced incomplete block designs with doubly transitive automorphism groups. J Combin Theory Ser A, 1973, 14: 221–247

    Article  MATH  MathSciNet  Google Scholar 

  54. Feit W. Some consequences of the classification of the finite simple groups. Proc Symp Pure Math, 1980, 37: 175–181

    MathSciNet  Google Scholar 

  55. Fried M D. On a conjecture of Schur. Michigen Math J, 1970, 17: 41–55

    Article  MATH  MathSciNet  Google Scholar 

  56. Fried M D. The field of definition of function fields and a problem in the reducibility of polynomials in two variables. Illinois J Math, 1973, 17: 128–146

    MATH  MathSciNet  Google Scholar 

  57. Fried M D. A theorem of Ritt and related diophantine problems. Crelles J, 1973, 264: 40–55

    Article  MATH  MathSciNet  Google Scholar 

  58. Fried M D. On Hilberts irreducibility theorem. JNT, 1974, 6: 211–232

    Article  MATH  MathSciNet  Google Scholar 

  59. Fried M D. On a theorem of MacCluer. Acta Arith, 1974, XXV: 122–127

    Google Scholar 

  60. Fried M D. Arithmetical properties of function fields (II): The generalized Schur problem. Acta Arith, 1974, XXV: 225–258

    MathSciNet  Google Scholar 

  61. Fried M D. Fields of definition of function fields and Hurwitz families and; Groups as Galois groups. Commun Algebra, 1977, 5: 17–82

    Article  MATH  MathSciNet  Google Scholar 

  62. Fried M D. Galois groups and Complex Multiplication. Trans Amer Math Soc, 1978, 235: 141–162

    Article  MATH  MathSciNet  Google Scholar 

  63. Fried M D. Exposition on an Arithmetic-Group theoretic connection via Riemanns Existence Theorem. Proc Symp Pure Math, 1980, 37: 571–601

    MathSciNet  Google Scholar 

  64. Fried M D. L-series on a Galois stratification. J Number Theory, 1986

  65. Fried M D. Irreducibility results for separated variables equations. J Pure Appl Algebra, 1987, 48: 9–22

    Article  MATH  MathSciNet  Google Scholar 

  66. Fried M D. Arithmetic of 3 and 4 branch point covers: a bridge provided by noncongruence subgroups of SL2(Z). Progress Math, 1990, 81: 77–117

    MathSciNet  Google Scholar 

  67. Fried M D. Global construction of general exceptional covers, with motivation for applications to coding. Contemp Math, 1994, 168: 69–100

    MathSciNet  Google Scholar 

  68. Fried M D. Enhanced review of J.P. Serres Topics in Galois Theory, with examples illustrating braid rigidity. Contemp Math, 1995, 186: 15–32

    MathSciNet  Google Scholar 

  69. Fried M D. Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture. Finite Fields Appl, 1995, 1: 326–359

    Article  MATH  MathSciNet  Google Scholar 

  70. Fried M D. Introduction to Modular Towers: Generalizing the relation between dihedral groups and modular curves. Contemp Math, 1995, 186: 111–171

    MathSciNet  Google Scholar 

  71. Fried M D. Separated variables polynomials and moduli spaces. In: Number Theory in Progress. Berlin-New York: Walter de Gruyter, 1999

    Google Scholar 

  72. Fried M D. Relating two genus 0 problems of John Thompson. Progress in Galois Theory, 2005, 51–85

  73. Fried M D. The place of exceptional covers among all diophantine relations. J Finite Fields, 2005, 11: 367–433

    Article  MATH  MathSciNet  Google Scholar 

  74. Fried M D. Should Journals compensate Referees? Notices Amer Math Soc, 2007, 25: 585–589

    Google Scholar 

  75. Fried M D. Algebraic Equations and Finite Simple Groups. Miami: University of Michigan, Department of Mathematics, 2008

    Google Scholar 

  76. Fried M D. Riemann’s Existence Theorem: An elementary approach to moduli. In preparation

  77. Fried M D. Paths that are classical generators of the punctured sphere. http://math.uci.edu/~mfried/deflistcov/classicalgens.pdf

  78. Fried M D. The genus 0 problem for rational functions. http://math.uci.edu/~mfried/deflist-cov/Genus0-Prob.html

  79. Fried M D. Alternating groups and moduli space lifting Invariants. Israel J Math, 2010, 179: 57–125

    Article  MATH  MathSciNet  Google Scholar 

  80. Fried M D, Guralnick R, Saxl J. Schur covers and carlitzs conjecture. Israel J Math, 1993, 82: 157–225

    Article  MATH  MathSciNet  Google Scholar 

  81. Fried M D, Gusić I. Schinzel’s Problem: Imprimitive covers and the monodromy method. Acta Arith, 2012, in press

  82. Fried M D, Jarden M. Field arithmetic. In: Ergebnisse der Mathematik III, vol. 11. Berlin: Springer-Verlag, 2004

    Google Scholar 

  83. Fried M D, MacRae R E. On the invariance of chains of fields. Illinois J Math, 1969, 13: 165–171

    MATH  MathSciNet  Google Scholar 

  84. Fried M D, Lidl R. On dickson polynomials and Rédei functions. In: Contributions to General Algebra 5. Vienna: Hölder-Pichler-Tempsky, 1987

    Google Scholar 

  85. Fried M D, Mezard A. Configuration Spaces for Wildly Ramified Covers. Providence, RI: Amer Math Soc, 2002, 353–376

    Google Scholar 

  86. Fried M D, Sacerdote G. Solving diophantine problems over all residue class fields of a number field. Ann of Math, 1976, 104: 203–233

    Article  MATH  MathSciNet  Google Scholar 

  87. Fried M D, Völklein H. The inverse Galois problem and rational points on moduli spaces. Math Ann, 1991, 290: 771–800

    Article  MATH  MathSciNet  Google Scholar 

  88. Fried M D, Völklein H. The embedding problem over an Hilbertian PAC field. Ann of Math, 1992, 135: 469–481

    Article  MATH  MathSciNet  Google Scholar 

  89. Fried M D, Whitley R. Effective Branch Cycle Computation. Preprint

  90. Fulton W. Fundamental Group of A Curve. Princeton: Princeton University Library, 1966

    Google Scholar 

  91. Garuti M. Prolongement de revêtements galoisiens en géométrie rigide. Comp Math, 1996, 104: 305–331

    MATH  MathSciNet  Google Scholar 

  92. Gorenstein D, Lyons R, Solomon R. The Classification of Finite Simple Groups. In: Mathematical Surveys and Monographs. Providence, RI: Amer Math Soc, 2006

    Google Scholar 

  93. Griffiths P. Periods of integrals on algebraic manifolds. Bull Amer Math Soc, 1970, 76: 228–296

    Article  MATH  MathSciNet  Google Scholar 

  94. Grothendieck A. Géométrie formelle et géométrie algébraique. Séminaire Bourbaki, 1958/59

  95. Guralnick R. Monodromy groups of coverings of curves. Galois groups and fundamental groups. Cambridge: Cambridge University Press, 2003

    Google Scholar 

  96. Guralnick R, Frohardt D, Magaard K. Genus 0 actions of groups of Lie rank 1. Proc Symp Pure Math, 2002, 70: 449–484

    MathSciNet  Google Scholar 

  97. Guralnick R, Magaard K. On the minimal degree of a permutation representation. J Algebra, 1998, 207: 127–145

    Article  MATH  MathSciNet  Google Scholar 

  98. Guralnick R, Müller P. Exceptional polynomials of affine type. J Algebra, 1997, 194: 429–454

    Article  MATH  MathSciNet  Google Scholar 

  99. Guralnick R, Müller P, Saxl J. The rational function analoque of a question of Schur and exceptionality of permutations representations. Memoirs Amer Math Soc, 2003

  100. Guralnick R, Shareshian J. Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points. Memoirs Amer Math Soc, 2007

  101. Guralnick R, Thompson J G. Finite groups of genus zero. J Algebra, 1990, 131: 303–341

    Article  MATH  MathSciNet  Google Scholar 

  102. Guralnick R, Tucker T J, Zieve M. Exceptional covers and bijections on rational points. ArXiv: 0511276v2

  103. Gusić I. Reducibility of f(x) − cf(y). Preprint, 2010

  104. Hall M. The Theory of Groups. Boston: MacMillan, 1963

    Google Scholar 

  105. Halmos P. I Have a Photographic Memory. Providence, RI: Amer Math Soc, 1987

    MATH  Google Scholar 

  106. Harbater D. Abhyankars conjecture on Galois groups over curves. Invent Math, 1994, 117: 1–25

    Article  MATH  MathSciNet  Google Scholar 

  107. Hales T. What is motivic measure? Bull Amer Math Soc, 2005, 42: 119–135

    Article  MATH  MathSciNet  Google Scholar 

  108. Hartshorne R. Algebraic Geometry. Berlin: Springer-Velag, 1977

    MATH  Google Scholar 

  109. Hilbert D. Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten. J Reine Angew Math, 1892, 110: 104–129

    Article  Google Scholar 

  110. Isaacs I M. Algebra, a Graduate Course. Florence: Brooks/Cole Publishing, 1994

    Google Scholar 

  111. Kanev V. Spectral curves, simple Lie algebras, and Prym-Tjurin varieties. Proc Sympos Pure Math, 1987, 49: 627–645

    MathSciNet  Google Scholar 

  112. Kiefe K. Sets definable over finite fields: Their zeta functions. Trans Amer Math Soc, 1976, 223: 45–59

    Article  MATH  MathSciNet  Google Scholar 

  113. Katz N M. Monodromy of families of curves: Applications of some results of Davenport-Lewis. Progress Math, 1981, 12: 171–195

    Google Scholar 

  114. Kriz I, Siegel P. Simple Groups at Play. Scientific American, 2008, 84–89

  115. Lang S. Algebra. Boston: Addison-Wesley, 1971

    Google Scholar 

  116. Lenstra H W, Zieve M. A family of exceptional polynomials in characteristic 3. London Math Soc Lecture, 1996, 233: 209–218

    MathSciNet  Google Scholar 

  117. LeVeque W J. On the equation ym = f(x). Acta Arith, 1964, 9: 209–219

    MATH  MathSciNet  Google Scholar 

  118. Lewis D J, Schinzel A. Quadratic diophantine equations with parameters. Acta Arith, 1980, 37: 133–141

    MATH  MathSciNet  Google Scholar 

  119. Lidl R, Mullen G L, Turnwald G. Dickson Polynomials. Essex: Longman Scientific, 1993

    MATH  Google Scholar 

  120. Liebeck M, Praeger C, Saxl J. The maximal factorizations of the finite simple groups and their automorphism Groups. Memoirs Amer Math Soc, 1990

  121. MacCluer C. On a conjecture of Davenport and Lewis concerning exceptional polynomials. Acta Arith, 1967, 12: 289–299

    MATH  MathSciNet  Google Scholar 

  122. Matthews R. Permutation polynomials over algebraic number fields. J Number Theory, 1984, 18: 249–260

    Article  MATH  MathSciNet  Google Scholar 

  123. Mazur B. Modular curves and the Eisenstein ideal. IHES Publ Math, 1977, 47: 33–186

    MATH  MathSciNet  Google Scholar 

  124. Merel L. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent Math, 1996, 124: 437–449

    Article  MATH  MathSciNet  Google Scholar 

  125. Mestre J F. Extensions régulières de ℚ(t) de groupe de Galois à n. J Algebra, 1990, 131: 483–495

    Article  MATH  MathSciNet  Google Scholar 

  126. Müller P. Primitive monodromy groups of polynomials. Contemp Math, 1995, 186: 385–401

    Google Scholar 

  127. Müller P. Reducibility behavior of polynomials with varying coefficients. Israel J Math, 1996, 94: 59–91

    Article  MATH  MathSciNet  Google Scholar 

  128. Müller P. Kronecker conjugacy of polynomials. Trans Amer Math Soc, 1998, 350: 1823–1850

    Article  MATH  MathSciNet  Google Scholar 

  129. Müller P. (A n, S n)-realizations by polynomials-on a question of Fried. Finite Fields Appl, 1998, 4: 465–468

    Article  MATH  MathSciNet  Google Scholar 

  130. Müller P. The Degree 8 Examples in Davenport’s Problem. Preprint, 2006

  131. Mumford D. The Red Book: Introduction to Algebraic Geometry. Preprint

  132. Mumford D. Curves and Their Jacobians. Ann Arbor: Ann Arbor UM Press, 1976

    Google Scholar 

  133. Nicaise J. Relative Motives and the Theory of Pseudo-finite Fields. Int Math Res, 2010, 1–69

  134. Pakovich F. Prime and composite Laurent polynomials. Bull Sci Math, 2009, 133: 693–732

    Article  MATH  MathSciNet  Google Scholar 

  135. Pakovich F. On the equation P(f) = Q(g), where P,Q are polynomials and f, g are entire functions. Amer J Math, 2010, 132

  136. Pakovich F. Algebraic curves P(x)−Q(y) = 0 and functional equations. Complex Variables Elliptic Equations, 2011, 14: 199–213

    Article  MathSciNet  Google Scholar 

  137. Picard E. Démonstration dun théorème général sur les fonctions uniformes liées par une relation algébrique. Acta Math, 1887, XI: 1–12

    Article  MathSciNet  Google Scholar 

  138. Raynaud M. Revêtements de la droite affine en caractèristique p > 0 et conjecture d A bhyankar. Invent Math, 1994, 116: 425–462

    Article  MATH  MathSciNet  Google Scholar 

  139. Ritt J F. Prime and composite polynomials. Trans Amer Math Soc, 1922, 23: 51–66

    Article  MATH  MathSciNet  Google Scholar 

  140. Schinzel A. Reducibility of Polynomials. Int Cong Math Nice, 1971, 491–496

  141. Schinzel A. Selected Topics on Polynomials. Ann Arbor: Arbor UM Press, 1982

    MATH  Google Scholar 

  142. Serre J P. Abelian ℓ-adic representations and elliptic curves. Wellesley: A K Peters, 1998

    Google Scholar 

  143. Serre J P. Relèvements dans à n. CR Acad Sci Serial I, 1990, 111: 478–482

    Google Scholar 

  144. Serre J P. Topics in Galois Theory. Sudbury: Bartlett and Jones Publishers, 1992

    MATH  Google Scholar 

  145. Siegel C L. Über einige Anwendungen diophantischer Approximationen. Abh Preus Akad Wiss Phys Math, 1929, 1: 14–67

    Google Scholar 

  146. Solomon R. A Brief History of the Classification of the Finite Simple Groups. Bull Amer Math Soc, 2001, 3: 315–352

    Article  Google Scholar 

  147. Springer G. Introduction to Riemann Surfaces. Boston: Addison-Wesley, 1957

    MATH  Google Scholar 

  148. Tverberg H. A remark on Ehrenfeucht’s criterion for the irreducibility of polynomials. Prace Mat, 1963/64, 8: 117–118

    MathSciNet  Google Scholar 

  149. Tverberg H. A Study in Irreducibility of Polynomials. Ph.D. Thesis, Univ Bergen, 1968

  150. Turnwald G. On Schur’s conjecture. J Austral Math Soc Ser A, 1995, 58: 312–357

    Article  MATH  MathSciNet  Google Scholar 

  151. van der Waerden B L. Die Zerlegungs- und Trägheitsgruppe als Permutationsgruppen. Math Ann, 1935, 111: 731–733

    Article  MathSciNet  Google Scholar 

  152. van der Poorten A J. The growth conditions for recurrence sequences. Unpublished

  153. Vetro F. Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type D d. Manuscripta Math, 2008, 125: 353–368

    Article  MATH  MathSciNet  Google Scholar 

  154. Vetro F. On Hurwitz spaces of coverings with one special fiber. Pacific J Math, 2009, 240: 383–398

    Article  MATH  MathSciNet  Google Scholar 

  155. Vojta P. Diophantine Approximation and Value Distribution Theory. Berlin: Springer-Verlag, 1987

    Google Scholar 

  156. Völklein H. Groups as Galois Groups. In: Cambridge Studies in Advance Mathematics. Cambridge: Cambridge University Press, 1996

    Google Scholar 

  157. Weil A. L’arithmetique sur les courbes algébriques. Acta Math, 1928, 52: 281–315

    Article  MathSciNet  Google Scholar 

  158. Wohlfahrt K. An extension of F. Klein’s level concept. Illinois J Math, 1964, 8: 529–535

    MATH  MathSciNet  Google Scholar 

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    Michael D. Fried

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Fried, M.D. Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification. Sci. China Math. 55, 1–72 (2012). https://doi.org/10.1007/s11425-011-4324-4

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