##
**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.

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)

### Keywords:

estimates for number of solutions; Thue equation; polynomial congruences; Thue-Mahler equations; Ramanujan-Nagell equations; \(S\)-unit equations; primitive solutions
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\textit{C. L. Stewart}, J. Am. Math. Soc. 4, No. 4, 793--835 (1991; Zbl 0744.11016)

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