##
**S-unit equations and their applications.**
*(English)*
Zbl 0658.10023

New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 110-174 (1988).

[For the entire collection see Zbl 0644.00005.]

During the recent time it turned out that large classes of decomposable form equations can be reduced to systems of S-unit equations of type \[ (1)\quad x_ 0+x_ 1+...+x_ n=0\quad in\quad S-units\quad x_ 0,...,x_ n. \] And although S-unit equations (mostly in the case \(n=2)\) were already considered by some authors, just the above fact gave the occasion to work out a whole theory of S-unit equations. Later on several other applications of these equations arose and finally it became an extremely powerful tool in diophantine problems.

The paper under review can be considered as a very useful up-to-date monograph on the theory and applications of S-unit equations. It can be used equally well by those who want to study this method, and by those who intend to develop this theory further and are looking for other applications.

After the introduction and describing the notation, in § 2 Theorem 1 contains J. H. Evertse’s finiteness theorem [Compos. Math. 53, 225- 244 (1984; Zbl 0547.10008)] on S-unit equations of type (1). In § 3 the two variable case is examined, that is equations of type \[ (2)\quad \alpha_ 1x_ 1+\alpha_ 2x_ 2=1\quad in\quad S-units\quad x_ 1,x_ 2 \] with fixed coefficients \(\alpha_ 1,\alpha_ 2\). Theorem 2 gives an upper bound for the number of solutions of (2) [see J. H. Evertse, Invent. Math. 75, 561-584 (1984; Zbl 0521.10015)]. Theorem 3 was proved by the authors during the conference in Durham [Invent. Math. 92, No.3, 461-477 (1988)]. It is a surprisingly sharp statement, namely that there are only finitely many equivalence classes of equations (2) with more than two solutions. Here the bound two is already best possible. Theorem 5 gives a somewhat weaker, but effective version of Theorem 3. Theorem 4 contains effective upper bounds for the heights of the solutions of (2) [see K. Györy, Comment. Math. Helv. 54, 583- 600 (1979; Zbl 0437.12004)]. § 4 contains the proofs of Theorems 1 and 2. In § 5 the rational integer case of Theorems 1-5 are formulated. This paragraph can be read separately from the rest of the paper and is intended for simpler applications.

§ 6-13 contain remarkable applications of S-unit equations. In § 6 a solution of a conjecture of Newman is given on the representation of numbers in the form \(2^{\alpha}3^{\beta}+2^{\tau}+3^{\delta}\) (\(\alpha\),\(\beta\),\(\tau\),\(\delta\geq 0\) integers). § 7 contains applications to finitely generated groups. In § 8,9, S-unit equations are applied to recurrence sequences and a slight generalization of the result of J. H. Evertse [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008)] is proved. In § 10, applications to arithmetical graphs [cf. K. Györy, Publ. Math. 27, 229-242 (1980; Zbl 0466.05047)] and to irreducibility of polynomials [cf. K. Györy, J. Number Theory 15, 164-181 (1982; Zbl 0509.12001)] are given.

In § 11 decomposable form equations are considered. For certain restricted classes of decomposable form equations Evertse and Györy gave bounds for the solutions themselves and also for the number of solutions. Moreover they gave criteria for the finiteness of the number of solutions of general decomposable form equations. For a detailed treatment of these results see also J. H. Evertse and K. Györy [New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 175-202 (1988; Zbl 0658.10022)].

In § 12 they formulate an effective theorem of K. Györy [Acta Arith. 23, 419-426 (1973; Zbl 0269.12001)] on integers with given discriminant and a result of J. H. Evertse and K. Györy [Acta Math. Hung. 51, No.3/4, 341-362 (1988; Zbl 0653.13005)] on the number of such elements.

Finally, § 13 contains a theorem of K. Nishioka [Compos. Math. 62, 53-61 (1987; Zbl 0615.10042)] on the algebraic independence of certain function values.

This paper will surely have a great influence on the further development of the theory of diophantine equations.

During the recent time it turned out that large classes of decomposable form equations can be reduced to systems of S-unit equations of type \[ (1)\quad x_ 0+x_ 1+...+x_ n=0\quad in\quad S-units\quad x_ 0,...,x_ n. \] And although S-unit equations (mostly in the case \(n=2)\) were already considered by some authors, just the above fact gave the occasion to work out a whole theory of S-unit equations. Later on several other applications of these equations arose and finally it became an extremely powerful tool in diophantine problems.

The paper under review can be considered as a very useful up-to-date monograph on the theory and applications of S-unit equations. It can be used equally well by those who want to study this method, and by those who intend to develop this theory further and are looking for other applications.

After the introduction and describing the notation, in § 2 Theorem 1 contains J. H. Evertse’s finiteness theorem [Compos. Math. 53, 225- 244 (1984; Zbl 0547.10008)] on S-unit equations of type (1). In § 3 the two variable case is examined, that is equations of type \[ (2)\quad \alpha_ 1x_ 1+\alpha_ 2x_ 2=1\quad in\quad S-units\quad x_ 1,x_ 2 \] with fixed coefficients \(\alpha_ 1,\alpha_ 2\). Theorem 2 gives an upper bound for the number of solutions of (2) [see J. H. Evertse, Invent. Math. 75, 561-584 (1984; Zbl 0521.10015)]. Theorem 3 was proved by the authors during the conference in Durham [Invent. Math. 92, No.3, 461-477 (1988)]. It is a surprisingly sharp statement, namely that there are only finitely many equivalence classes of equations (2) with more than two solutions. Here the bound two is already best possible. Theorem 5 gives a somewhat weaker, but effective version of Theorem 3. Theorem 4 contains effective upper bounds for the heights of the solutions of (2) [see K. Györy, Comment. Math. Helv. 54, 583- 600 (1979; Zbl 0437.12004)]. § 4 contains the proofs of Theorems 1 and 2. In § 5 the rational integer case of Theorems 1-5 are formulated. This paragraph can be read separately from the rest of the paper and is intended for simpler applications.

§ 6-13 contain remarkable applications of S-unit equations. In § 6 a solution of a conjecture of Newman is given on the representation of numbers in the form \(2^{\alpha}3^{\beta}+2^{\tau}+3^{\delta}\) (\(\alpha\),\(\beta\),\(\tau\),\(\delta\geq 0\) integers). § 7 contains applications to finitely generated groups. In § 8,9, S-unit equations are applied to recurrence sequences and a slight generalization of the result of J. H. Evertse [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008)] is proved. In § 10, applications to arithmetical graphs [cf. K. Györy, Publ. Math. 27, 229-242 (1980; Zbl 0466.05047)] and to irreducibility of polynomials [cf. K. Györy, J. Number Theory 15, 164-181 (1982; Zbl 0509.12001)] are given.

In § 11 decomposable form equations are considered. For certain restricted classes of decomposable form equations Evertse and Györy gave bounds for the solutions themselves and also for the number of solutions. Moreover they gave criteria for the finiteness of the number of solutions of general decomposable form equations. For a detailed treatment of these results see also J. H. Evertse and K. Györy [New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 175-202 (1988; Zbl 0658.10022)].

In § 12 they formulate an effective theorem of K. Györy [Acta Arith. 23, 419-426 (1973; Zbl 0269.12001)] on integers with given discriminant and a result of J. H. Evertse and K. Györy [Acta Math. Hung. 51, No.3/4, 341-362 (1988; Zbl 0653.13005)] on the number of such elements.

Finally, § 13 contains a theorem of K. Nishioka [Compos. Math. 62, 53-61 (1987; Zbl 0615.10042)] on the algebraic independence of certain function values.

This paper will surely have a great influence on the further development of the theory of diophantine equations.

Reviewer: I.Gaál

### MSC:

11D57 | Multiplicative and norm form equations |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11D61 | Exponential Diophantine equations |