9.3 Proof of Proposition 9.2
9.3.4 Concluding the proof of Proposition 9.2
Now Proposition 9.11 also provides upper bounds for the heights of elements of the setC1. So for any (x0, y0)∈ C1 we have
degx0,degy0 ≤exp
N6(2d)expO(r+s)(h+ 1)4s , and
h(x0), h(y0)≤exp exp
N12(2d)expO(r+s)(h+ 1)8s , which by Lemma 6.4 concludes the proof of our Proposition 9.2.
Bibliography
[1] M. Aschenbrenner, Ideal membership in polynomial rings over the integers, J.
Amer. Math. Soc.,17 (2004), 407–442.
[2] A. Baker, Contributions to the theory of Diophantine equations I. On the repre-sentation of integers by binary forms, Philos. Trans. Roy. Soc. London Ser. A, 263 (1967/68), 173–191.
[3] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc., 65 (1969), 439–444.
[4] A. Baker,Transcendental number theory, Cambridge University Press, London-New York, 1975.
[5] A. Bakerand G. W¨ustholz,Logarithmic forms and Diophantine geometry, vol. 9 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2007.
[6] A. B´erczes, Effective results for division points on curves inG2m,J. Th´eor. Nombres Bordeaux,27 (2015), 405–437.
[7] A. B´erczes, Effective results for unit points on curves over finitely generated do-mains, Math. Proc. Cambridge Phil. Soc., 158 (2015), 331–353.
[8] A. B´erczes,J.-H. EvertseandK. Gy˝ory, Effective results for linear equations in two unknowns from a multiplicative division group,Acta Arith.,136(2009), 331–349.
[9] A. B´erczes, J.-H. Evertse and K. Gy˝ory, Effective results for hyper- and su-perelliptic equations over number fields, Publ. Math. Debrecen,82 (2013), 727–756.
[10] A. B´erczes,J.-H. EvertseandK. Gy˝ory, Effective results for Diophantine equa-tions over finitely generated domains, Acta Arith.,163 (2014), 71–100.
[11] A. B´erczes, J.-H. Evertse, K. Gy˝ory and C. Pontreau, Effective results for points on certain subvarieties of tori, Math. Proc. Cambridge Phil. Soc., 147 (2009), 69–94.
[12] F. Beukers and H. P. Schlickewei, The equation x+y= 1 in finitely generated groups, Acta Arith., 78 (1996), 189–199.
[13] F. Beukers and C. J. Smyth, Cyclotomic points on curves, in: Number theory for the millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 67–85.
[14] F. Beukers and D. Zagier, Lower bounds of heights of points on hypersurfaces, Acta Arith., 79 (1997), 103–111.
[15] E. Bombieri, Effective Diophantine approximation onGm,Ann. Scuola Norm. Sup.
Pisa Cl. Sci. (4), 20 (1993), 61–89.
[16] E. Bombieri and P. B. Cohen, Effective Diophantine approximation on GM. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),24 (1997), 205–225.
[17] E. Bombieri and P. B. Cohen, An elementary approach to effective Diophantine approximation on Gm, in: Number theory and algebraic geometry, Cambridge Univ.
Press, Cambridge, 2003, pp. 41–62.
[18] E. Bombieri and W. Gubler, Heights in Diophantine geometry, Cambridge Uni-versity Press, Cambridge, 2006.
[19] E. Bombieri and U. Zannier, Algebraic points on subvarieties of Gnm, Internat.
Math. Res. Notices, (1995), 333–347.
[20] I. Borosh, M. Flahive, D. Rubin and B. Treybig, A sharp bound for solutions of linear Diophantine equations, Proc. Amer. Math. Soc., 105 (1989), 844–846.
[21] B. Brindza, OnS-integral solutions of the equationym =f(x),Acta Math. Hungar., 44 (1984), 133–139.
[22] B. Brindza, On the equationf(x) = ym over finitely generated domains,Acta Math.
Hungar., 53 (1989), 377–383.
[23] B. Brindza, The Catalan equation over finitely generated integral domains, Publ.
Math. Debrecen,42 (1993), 193–198.
[24] B. Brindzaand A. Pint´´ er, On equal values of binary forms over finitely generated fields, Publ. Math. Debrecen, 46 (1995), 339–347.
[25] W. D. Brownawell andD. W. Masser, Vanishing sums in function fields, Math.
Proc. Cambridge Philos. Soc., 100 (1986), 427–434.
[26] Y. Bugeaud, Bornes effectives pour les solutions des ´equations en S-unit´es et des
´
equations de Thue-Mahler, J. Number Theory, 71 (1998), 227–244.
[27] Y. BugeaudandK. Gy˝ory, Bounds for the solutions of unit equations,Acta Arith., 74 (1996), 67–80.
[28] J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1959.
[29] J. Coates, An effective p-adic analogue of a theorem of Thue, Acta Arith., 15 (1968/69), 279–305.
[30] J.-H. Evertse, Points on subvarieties of tori, in: A panorama of number theory or the view from Baker’s garden (Z¨urich, 1999), Cambridge Univ. Press, 2002, pp.
214–230.
[31] J.-H. Evertse and K. Gy˝ory, Discriminant Equations in Diophantine Number Theory, Cambridge University Press, to appear.
[32] J.-H. Evertse and K. Gy˝ory, Effective results for unit equations over finitely generated integral domains, Math. Proc. Camb. Phil. Soc., 154 (2013), 351–380.
[33] J.-H. EvertseandK. Gy˝ory,Unit Equation in Diophantine Number Theory, Cam-bridge University Press, 2015.
[34] J.-H. Evertse, H. P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group,Ann. of Math. (2), 155(2002), 807–836.
[35] E. Friedman, Analytic formulas for the regulator of a number field., Invent. Math., 98 (1989), 599–622.
[36] K. Gy˝ory, Sur l’irr´eductibilit´e d’une classe des polynˆomes. II,Publ. Math. Debrecen, 19 (1972), 293–326.
[37] K. Gy˝ory, Sur les polynˆomes `a coefficients entiers et de dicriminant donn´e II,Publ.
Math. Debrecen,21 (1974), 125–144.
[38] K. Gy˝ory, On the number of solutions of linear equations in units of an algebraic number field, Comment. Math. Helv.,54 (1979), 583–600.
[39] K. Gy˝ory, Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains, Acta Math. Hungar., 42 (1983), 45–
80.
[40] K. Gy˝ory, Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely generated domains, J. Reine Angew. Math., 346 (1984), 54–100.
[41] K. Gy˝oryandK. Yu, Bounds for the solutions ofS-unit equations and decomposable form equations, Acta Arith., 123 (2006), 9–41.
[42] G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann., 95 (1926), 736–788.
[43] M. HindryandJ. H. Silverman,Diophantine geometry, vol. 201 ofGraduate Texts in Mathematics, Springer-Verlag, New York, 2000.
[44] S. Lang, Integral points on curves, Inst. Hautes ´Etudes Sci. Publ. Math., (1960), 27–43.
[45] S. Lang,Diophantine geometry, Interscience Tracts in Pure and Applied Mathemat-ics, No. 11, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962.
[46] S. Lang, Division points on curves, Ann. Mat. Pura Appl. (4),70 (1965), 229–234.
[47] S. Lang, Report on diophantine approximations,Bull. Soc. Math. France,93(1965), 177–192.
[48] S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983.
[49] M. Laurent, Equations diophantiennes exponentielles, Invent. Math., 78 (1984), 299–327.
[50] W. LeVeque, On the equation ym =f(x),Acta Arith., 9 (1964), 209–219.
[51] P. Liardet, Sur une conjecture de Serge Lang, C. R. Acad. Sci. Paris S´er. A, 279 (1974), 435–437.
[52] P. Liardet, Sur une conjecture de Serge Lang, in: Journ´ees Arithm´etiques de Bor-deaux (Conf., Univ. BorBor-deaux, BorBor-deaux, 1974), Soc. Math. France, Paris, 1975, pp.
187–210. Ast´erisque, Nos. 24–25.
[53] S. Louboutin, Explicit bounds for residues of Dedekind zeta functions, values of L-functions at s = 1, and relative class numbers, J. Number Theorey, 85 (2000), 263–282.
[54] K. Mahler, Zur Approximation algebraischer Zahlen. I, Math. Ann., 107 (1933), 691–730.
[55] R. C. Mason, Diophantine equations over function fields, Cambridge University Press, 1984.
[56] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II (translated from Russian), Izv. Math., 64 (2000), 1217–1269.
[57] C. J. Parry, The p-adic generalisation of the Thue-Siegel theorem, Acta Math., 83 (1950), 1–100.
[58] C. Pontreau, A Mordell-Lang plus Bogomolov type result for curves in G2m, Monatsh. Math., 157 (2009), 267–281.
[59] C. Pontreau, Petits points d’une surface, Canad. J. Math., 61 (2009), 1118–1150.
[60] B. Poonen, Mordell-Lang plus Bogomolov, Invent. Math.,137 (1999), 413–425.
[61] G. R´emond, D´ecompte dans une conjecture de Lang, Invent. Math., 142 (2000), 513–545.
[62] G. R´emond, Sur les sous-vari´et´es des tores, Compositio Math.,134 (2002), 337–366.
[63] G. R´emond, Approximation diophantienne sur les vari´et´es semi-ab´eliennes,Ann. Sci.
Ecole Norm. Sup. (4),´ 36 (2003), 191–212.
[64] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64–94.
[65] A. SchinzelandR. Tijdeman, On the equationym =P(x),Acta Arith.,31(1976), 199–204.
[66] H. P. Schlickewei, Lower bounds for heights on finitely generated groups,Monatsh.
Math., 123 (1997), 171–178.
[67] W. M. Schmidt, Thue’s equation over function fields, J. Austral. Math. Soc. Ser.
A.,25 (1978), 385–442.
[68] W. M. Schmidt, Diophantine approximation, vol. 785 of Lecture Notes in Mathe-matics, Springer, Berlin, 1980.
[69] W. M. Schmidt, Heights of points on subvarieties ofGnm, in: Number theory (Paris, 1993–1994), Cambridge Univ. Press, Cambridge, 1996, vol. 235 ofLondon Math. Soc.
Lecture Note Ser., pp. 157–187.
[70] C. Siegel, Approximation algebraischer Zahlen,Math. Z.,10 (1921), 173–213.
[71] V. G. Sprindˇzuk and S. V. Kotov, An effective analysis of the Thue-Mahler equation in relative fields (Russian), Dokl. Akad. Nauk. BSSR, 17 (1973), 393–395, 477.
[72] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23 (1974), 135–152.
[73] A. Thue, ¨Uber Ann¨aherungswerte algebraischer Zahlen,J. Reine Angew. Math.,135 (1909), 284–305.
[74] L. A. Trelina, S-integral solutions of Diophantine equations of hyperbolic type (in Russian), Dokl. Akad. Nauk. BSSR, 22 (1978), 881–884;955.
[75] J. V´egs˝o, On superelliptic equations, Publ. Math. Debrecen, 44 (1994), 183–187.
[76] P. Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith.,74 (1996), 81–95.
[77] M. Waldschmidt,Diophantine approximation on linear algebraic groups, Springer-Verlag, 2000.
[78] K. Yu, P-adic logarithmic forms and group varieties. III, Forum Math., 19 (2007), 187–280.
[79] S. Zhang, Positive line bundles on arithmetic varieties,J. Amer. Math. Soc.,8(1995), 187–221.