## On Thue’s equation.(English)Zbl 0614.10018

In this very interesting paper the authors consider the number $$N(F,h)$$ of solutions of Thue’s equation $$F(x,y)=h$$ in integers $$x,y\in\mathbb{Z}$$ with $$(x,y)=1$$. Here, $$F\in\mathbb{Z}[x,y]$$ is a binary form of degree $$r\ge 3$$ and h is fixed. In 1983 J. H. Evertse was the first to give a bound for $$N(F,h)$$ depending only on $$r$$ and $$h$$, but independent of the coefficients of $$F$$. The dependence on $$r$$ is of the form $$\exp(r^2)$$. In the present paper the authors give a vast improvement of Evertse’s theorem. They show $$N(F,h)<c_1r^{1+t}$$, where $$t$$ is the number of distinct prime divisors of $$h$$ and $$c_1$$ is some computable constant which is $$<215$$ if $$r$$ is big enough. In case $$h=1$$ this bound is optimal in its dependence on $$r$$, since $$x^r+(y-x)(y-2x) \cdots (y-rx)=1$$ obviously has at least $$r$$ solutions.
The proof starts with the case $$h=1$$ and then, by induction on $$t$$, is extended to general $$h$$. To bound the number of large solutions the classical Thue-Siegel method is used. For the small solutions the authors first split the original problem in a union of Thue equations with large discriminant and then use some ingenious proximity arguments to finish the proof.

### MSC:

 11D45 Counting solutions of Diophantine equations 11D59 Thue-Mahler equations
Full Text:

### References:

 [1] Bombieri, E., Mueller, J.: On effective measures of irrationality for $$\sqrt[r]{{\frac{a}{b}}}$$ and related numbers. J. Reine Angew. Math.342, 173-196 (1983) · Zbl 0516.10024 [2] Evertse, J.-H.: Upper bounds for the number of solutions of diophantine equations. Math. Centrum. Amsterdam, pp. 1-127 (1983) · Zbl 0517.10016 [3] Evertse, J.-H.: On the representation of integers by binary cubic forms of positive discriminant. Invent. Math.73, 117-138 (1983) · Zbl 0506.10013 [4] Hooley, C.: On the representations of numbers by binary cubic forms. Glasgow Math. J.27, 95-98 (1985) · Zbl 0577.10021 [5] Lewis, D., Mahler, K.: Representation of integers by binary forms. Acta Arith.6, 333-363 (1961) · Zbl 0102.03601 [6] Mahler, K.: An inequality for the discriminant of a polynomial. Michigan Math. J.11, 257-262 (1964) · Zbl 0135.01702 [7] Mahler, K.: On Thue’s theorem. Austr. Nat. Univ. Res. Rep. 24 (1982). Math. Scand.55, 188-200 (1984) · Zbl 0544.10014 [8] Siegel, C.L.: Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), N{$$\deg$$} 1. Gesammelte Abhandlungen, Bd. 1, pp. 209-266. Berlin Heidelberg New York: Springer-Verlag 1966 · JFM 56.0180.05 [9] Silverman, J.H.: Representation of integers by binary forms and the rank of the Mordell-Weil group. Invent. Math.74, 218-292 (1983) · Zbl 0525.14012
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.