## On the number of solutions of polynomial congruences and Thue equations.(English)Zbl 0744.11016

In the present paper the author gives new and deep estimates for the number of solutions of some classical diophantine equations like Thue and Thue-Mahler equations and Ramanujan-Nagell equations, improving many earlier results.
Let $$F(X,Y)$$ be a binary form with rational integer coefficients, of degree $$r\geq 3$$, content 1 and non-zero discriminant. Further, let $$h$$ be a non-zero integer and $$\varepsilon$$ a positive real number. In Theorem 1 it is shown, that the number of coprime solutions $$(x,y)=1$$ of the Thue equation $F(x,y)=h(1)$ is at most $$2800(1+1/(8\varepsilon r))r^{1+\omega(g)}$$ where $$g$$ is a suitable divisor of $$h$$. For most integers $$h$$ this theorem improves on the result of E. Bombieri and W. M. Schmidt [Invent. Math. 88, 69-81 (1987; Zbl 0614.10018)]. The proof depends on the Thue Siegel principle, as the arguments of Bombieri and Schmidt. Moreover, the proof involves upper bounds for the number of solutions of polynomial congruences, that are in general best possible (Theorem 2).
For each binary form $$F(X,Y)$$ with non-zero discriminant and degree $$r\geq 3$$ denote by $$\nu(F)$$ the largest integer $$k$$, such that (1) has at least $$k$$ primitive solutions for arbitrary large $$h$$. Further, let $$\nu^*(r)$$ be the supremum of $$\nu(F)$$ over all binary forms $$F$$ with non-zero discriminant and degree $$r$$. Theorem 3 establishes lower bounds for $$\nu^*(r)$$ that are between 2 and 12 depending on the congruence behavior of $$r$$ modulo 6.
Theorem 4 deals with Thue-Mahler equations of type $F(x,y)=hp_ 1^{k_ 1}\dots p_ t^{k_ t}(2)$ where $$F$$ is a form with non-zero discriminant, degree $$r$$ and content 1 as before, $$p_ 1,\dots,p_ t$$ are distinct primes, $$h$$ coprime to $$p_ 1,\dots,p_ t$$ and we consider the solutions in integers with $$(x,y)=1$$, $$k_ 1,\dots,k_ t\geq 0$$. The author shows that if $$h$$ is sufficiently large, then the number of solutions of (2) is at most $$4r^ \omega(h)$$. Moreover, if $$h$$ is larger than an effectively computable lower bound, then the number of solutions is $$\leq 2(t+1)r^{3+\omega(h)}$$. The theorem improves the result of E. Bombieri [Lect. Notes Math. 1290, 213-243 (1987; Zbl 0646.10009)].
Finally, in Theorem 5 Ramanujan-Nagell equations of type $f(x)=hp^{k_ 1}_ 1\dots p_ t^{k-t}(3)$ are considered, where $$f$$ is a polynomial with integer coefficients, content 1, degree $$r$$ ($$\geq 2$$) and non-zero discriminant, $$p_ 1,\dots,p_ t$$ distinct primes and $$h$$ is an integer, coprime to $$p_ 1,\dots,p_ t$$. It is shown, that if $$h$$ is sufficiently large, then the number of solutions of (3) in integers $$x$$; $$k_ 1,\dots,k_ t\geq 0$$ is at most $$2r^{\omega(h)}$$. Further, if $$h$$ is larger than an effectively computable constant, then the number of solutions is at most $$(t+1)r^{2+\omega(h)}$$. The proof of the last two theorems uses the results of J.-H. Evertse, K. Györy, C. L. Stewart and R. Tijdeman [Invent. Math. 92, 461-477 (1988; Zbl 0662.10012)] on the number of solutions of $$S$$-unit equations.
Reviewer: I.Gaál (Debrecen)

### MSC:

 11D41 Higher degree equations; Fermat’s equation 11D75 Diophantine inequalities

### Citations:

Zbl 0614.10018; Zbl 0646.10009; Zbl 0662.10012
Full Text:

### References:

  Yvette Amice, Les nombres \?-adiques, Presses Universitaires de France, Paris, 1975 (French). Préface de Ch. Pisot; Collection SUP: Le Mathématicien, No. 14. · Zbl 0313.12104  Enrico Bombieri, On the Thue-Siegel-Dyson theorem, Acta Math. 148 (1982), 255 – 296. · Zbl 0505.10015  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.  E. Bombieri and J. Mueller, On effective measures of irrationality for $$\sqrt[r]{{\frac{a} {b}}}$$ and related numbers, J. Reine Angew. Math. 342 (1983), 173-196. · Zbl 0516.10024  E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. 88 (1987), no. 1, 69 – 81. · Zbl 0614.10018  N. G. de Bruijn, On the number of positive integers \le \? and free of prime factors >\?, Nederl. Acad. Wetensch. Proc. Ser. A. 54 (1951), 50 – 60. · Zbl 0042.04204  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. · Zbl 0493.12017  S. Chowla, Contributions to the analytic theory of numbers. II, J. Indian Math. Soc. 20 (1933), 120-128. · Zbl 0009.34002  P. Erdös and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), 134-139. · JFM 64.0116.01  P. Erdös, C. L. Stewart, and R. Tijdeman, Some Diophantine equations with many solutions, Compositio Math. 66 (1988), no. 1, 37 – 56. · Zbl 0639.10014  Jan-Hendrik Evertse, On the equation \?\?$$^{n}$$-\?\?$$^{n}$$=\?, Compositio Math. 47 (1982), no. 3, 289 – 315. · Zbl 0498.10014  -, Upper bounds for the numbers of solutions of diophantine equations, M.C.-Tract 168, Centre of Mathematics and Computer Science, Amsterdam, 1983.  J.-H. Evertse, On equations in \?-units and the Thue-Mahler equation, Invent. Math. 75 (1984), no. 3, 561 – 584. · Zbl 0521.10015  J.-H. Evertse and K. Győry, Thue-Mahler equations with a small number of solutions, J. Reine Angew. Math. 399 (1989), 60 – 80. · Zbl 0675.10009  J.-H. Evertse and K. Győry, Effective finiteness results for binary forms with given discriminant, Compositio Math. 79 (1991), no. 2, 169 – 204. · Zbl 0746.11020  -, Thue inequalities with a small number of solutions (to appear).  J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, On \?-unit equations in two unknowns, Invent. Math. 92 (1988), no. 3, 461 – 477. · Zbl 0662.10012  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. · Zbl 0423.10001  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.  Serge Lang, Integral points on curves, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27 – 43. · Zbl 0112.13402  D. J. Lewis and K. Mahler, On the representation of integers by binary forms, Acta Arith. 6 (1960/1961), 333 – 363. · Zbl 0102.03601  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. · Zbl 0474.10030  Kurt Mahler, Zur Approximation algebraischer Zahlen. II, Math. Ann. 108 (1933), no. 1, 37 – 55 (German). · Zbl 0006.15604  -, On the lattice points on curves of genus 1, Proc. London Math. Soc. (2) 39 (1935), 431-466. · JFM 61.0146.02  K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257 – 262. · Zbl 0135.01702  -, On Thue’s theorem, Math. Scand. 55 (1984), 188-200. · Zbl 0544.10014  L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. · Zbl 0188.34503  P. Moree and C. L. Stewart, Some Ramanujan-Nagell equations with many solutions, Indag. Math. (N.S.) 1 (1990), no. 4, 465 – 472. · Zbl 0718.11011  J. Mueller and W. M. Schmidt, Thue’s equation and a conjecture of Siegel, Acta Math. 160 (1988), no. 3-4, 207 – 247. · Zbl 0655.10016  T. Nagell, Généralisation d’un theórème de Tchebicheff, J. Math. 8 (1921), 343-356.  Morris Newman, Integral matrices, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. · Zbl 0254.15009  O. Ore, Anzahl der Wurzeln höherer Kongruenzen., Norsk Matematisk Tidsskrift, 3 Aagang, Kristiana (1921), 343-356. · JFM 48.1163.02  Gyula Sándor, Uber die Anzahl der Lösungen einer Kongruenz, Acta Math. 87 (1952), 13 – 16 (German). · Zbl 0046.26605  Wolfgang M. Schmidt, Thue equations with few coefficients, Trans. Amer. Math. Soc. 303 (1987), no. 1, 241 – 255. · Zbl 0634.10017  Carl Ludwig Siegel, Die Gleichung \?\?$$^{n}$$ – \?\?$$^{n}$$=\?, Math. Ann. 114 (1937), no. 1, 57 – 68 (German). · Zbl 0015.38902  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. · Zbl 0525.14012  Joseph H. Silverman, Integer points on curves of genus 1, J. London Math. Soc. (2) 28 (1983), no. 1, 1 – 7. · Zbl 0487.10015  Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.