On the number of solutions of polynomial congruences and Thue equations

On the number of solutions of polynomial congruences and Thue equations
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J. Amer. Math. Soc. 4 (1991), 793-835

DOI: https://doi.org/10.1090/S0894-0347-1991-1119199-X

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References

Yvette Amice, Les nombres $p$-adiques, Collection SUP: “Le Mathématicien”, vol. 14, Presses Universitaires de France, Paris, 1975 (French). Préface de Ch. Pisot. MR 0447195
Enrico Bombieri, On the Thue-Siegel-Dyson theorem, Acta Math. 148 (1982), 255–296. MR 666113, DOI 10.1007/BF02392731
E. Bombieri, On the Thue-Mahler equation, Diophantine approximation and transcendence theory (Bonn, 1985) Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, pp. 213–243. MR 927563, DOI 10.1007/BFb0078711

On effective measures of irrationality for

and related numbers

342

E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. 88 (1987), no. 1, 69–81. MR 877007, DOI 10.1007/BF01405092
N. G. de Bruijn, On the number of positive integers $\leq x$ and free of prime factors $>y$, Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50–60. MR 0046375
J. H. H. Chalk and R. A. Smith, Sándor’s theorem on polynomial congruences and Hensel’s lemma, C. R. Math. Rep. Acad. Sci. Canada 4 (1982), no. 1, 49–54. MR 646697

Contributions to the analytic theory of numbers

20

On the number of integers which can be represented by a binary form

13

P. Erdös, C. L. Stewart, and R. Tijdeman, Some Diophantine equations with many solutions, Compositio Math. 66 (1988), no. 1, 37–56. MR 937987
Jan-Hendrik Evertse, On the equation $ax^{n}-by^{n}=c$, Compositio Math. 47 (1982), no. 3, 289–315. MR 681611

Upper bounds for the numbers of solutions of diophantine equations

168

J.-H. Evertse, On equations in $S$-units and the Thue-Mahler equation, Invent. Math. 75 (1984), no. 3, 561–584. MR 735341, DOI 10.1007/BF01388644
J.-H. Evertse and K. Győry, Thue-Mahler equations with a small number of solutions, J. Reine Angew. Math. 399 (1989), 60–80. MR 1004133
J.-H. Evertse and K. Győry, Effective finiteness results for binary forms with given discriminant, Compositio Math. 79 (1991), no. 2, 169–204. MR 1117339

Thue inequalities with a small number of solutions

J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, On $S$-unit equations in two unknowns, Invent. Math. 92 (1988), no. 3, 461–477. MR 939471, DOI 10.1007/BF01393743
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
M. N. Huxley, A note on polynomial congruences, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 193–196. MR 637347
Serge Lang, Integral points on curves, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27–43. MR 130219
D. J. Lewis and K. Mahler, On the representation of integers by binary forms, Acta Arith. 6 (1960/61), 333–363. MR 120195, DOI 10.4064/aa-6-3-333-363
John H. Loxton and Robert A. Smith, On Hua’s estimate for exponential sums, J. London Math. Soc. (2) 26 (1982), no. 1, 15–20. MR 667239, DOI 10.1112/jlms/s2-26.1.15
Kurt Mahler, Zur Approximation algebraischer Zahlen. II, Math. Ann. 108 (1933), no. 1, 37–55 (German). MR 1512833, DOI 10.1007/BF01452821

On the lattice points on curves of genus

39

K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262. MR 166188

On Thue’s theorem

55

L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
P. Moree and C. L. Stewart, Some Ramanujan-Nagell equations with many solutions, Indag. Math. (N.S.) 1 (1990), no. 4, 465–472. MR 1106093, DOI 10.1016/0019-3577(90)90014-E
J. Mueller and W. M. Schmidt, Thue’s equation and a conjecture of Siegel, Acta Math. 160 (1988), no. 3-4, 207–247. MR 945012, DOI 10.1007/BF02392276

Généralisation d’un theórème de Tchebicheff

8

Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283

Anzahl der Wurzeln höherer Kongruenzen

Gyula Sándor, Uber die Anzahl der Lösungen einer Kongruenz, Acta Math. 87 (1952), 13–16 (German). MR 47679, DOI 10.1007/BF02392280
Wolfgang M. Schmidt, Thue equations with few coefficients, Trans. Amer. Math. Soc. 303 (1987), no. 1, 241–255. MR 896020, DOI 10.1090/S0002-9947-1987-0896020-1
Carl Ludwig Siegel, Die Gleichung $ax^n$–$by^n=c$, Math. Ann. 114 (1937), no. 1, 57–68 (German). MR 1513124, DOI 10.1007/BF01594162
J. H. Silverman, Representations of integers by binary forms and the rank of the Mordell-Weil group, Invent. Math. 74 (1983), no. 2, 281–292. MR 723218, DOI 10.1007/BF01394317
Joseph H. Silverman, Integer points on curves of genus $1$, J. London Math. Soc. (2) 28 (1983), no. 1, 1–7. MR 703458, DOI 10.1112/jlms/s2-28.1.1
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8