The number of solutions of decomposable form equations.

*(English)*Zbl 0851.11019Decomposable form equations, being the common generalization of several types of classical diophantine equations (among others Thue equations, norm form equations, discriminant form and index form equations) have an extensive literature. Using his famous subspace theorem, in 1972 W. M. Schmidt [Ann. Math., II. Ser. 96, 526-551 (1972; Zbl 0245.10008)] proved the finiteness of the number of solutions of nondegenerate norm form equations. In 1977 H. P. Schlickewei [J. Number Theory 9, 370-380 (1977; Zbl 0365.10016)], extended Schmidt’s result to \(p\)-adic norm form equations, and in 1984 M. Laurent [Invent. Math. 78, 299-327 (1984; Zbl 0554.10009)] to norm form equations over number fields. The first finiteness results on decomposable form equations were obtained by the author and K. Györy in 1988 [Acta Arith. 50, 357-379 (1988; Zbl 0595.10013)]. In 1989 the author, the reviewer and K. Györy [Arch. Math. 52, 337-353 (1989; Zbl 0671.10013)] showed the existence of a uniform bound \(C\) (depending only on the number of variables, the \(S\)-unit group involved, and the splitting field) such that any nondegenerate decomposable form equation (with the above parameters) can have at most \(C\) cosets of solutions. These results applied to Schmidt’s subspace theorem and its \(p\)-adic generalization given by Schlickewei.

In 1989 W. M. Schmidt [Compos. Math. 69, 121-173 (1989; Zbl 0683.10027)] gave a quantitative version of his subspace theorem which enabled him to derive an explicit upper bound for the number of solutions of norm form equations with a non-degenerate module. H. P. Schlickewei [Compos. Math. 82, 245-273 (1992; Zbl 0751.11033)] extended the quantitative subspace theorem to the number field case and obtained also an explicit upper bound for the number of solutions of \(S\)-unit equations over number fields. In 1993 K. Györy [Publ. Math. 42, 65-101 (1993; Zbl 0792.11004)] applied this result to give an explicit bound for the number of cosets of solutions of arbitrary decomposable form equations, and, extending the results of Schmidt and Schlickewei on norm form equations, derived also a bound for the number of families of solutions of possibly degenerate decomposable form equations.

In the present paper the author considers the number of non-degenerate solutions of possibly degenerate decomposable form equations. The only restrictive condition on the linear factors of the decomposable form equation is the following: for every proper nonempty subset of its linear factors there exist algebraic coefficients which can be used to make an identically zero linear combination of these linear factors without vanishing subsums. The bounds derived for the number of solutions improve the above mentioned quantitative results on the number of solutions of norm form and decomposable form equations. The author also derives an improvement of Schlickewei’s upper bound on the number of solutions of \(S\)-unit equations. The main tool of the proofs is an improvement of the quantitative subspace theorem given by the author [An improvement of the quantitative subspace theorem, Compos. Math. 101, No. 3, 225-311 (1996)].

In 1989 W. M. Schmidt [Compos. Math. 69, 121-173 (1989; Zbl 0683.10027)] gave a quantitative version of his subspace theorem which enabled him to derive an explicit upper bound for the number of solutions of norm form equations with a non-degenerate module. H. P. Schlickewei [Compos. Math. 82, 245-273 (1992; Zbl 0751.11033)] extended the quantitative subspace theorem to the number field case and obtained also an explicit upper bound for the number of solutions of \(S\)-unit equations over number fields. In 1993 K. Györy [Publ. Math. 42, 65-101 (1993; Zbl 0792.11004)] applied this result to give an explicit bound for the number of cosets of solutions of arbitrary decomposable form equations, and, extending the results of Schmidt and Schlickewei on norm form equations, derived also a bound for the number of families of solutions of possibly degenerate decomposable form equations.

In the present paper the author considers the number of non-degenerate solutions of possibly degenerate decomposable form equations. The only restrictive condition on the linear factors of the decomposable form equation is the following: for every proper nonempty subset of its linear factors there exist algebraic coefficients which can be used to make an identically zero linear combination of these linear factors without vanishing subsums. The bounds derived for the number of solutions improve the above mentioned quantitative results on the number of solutions of norm form and decomposable form equations. The author also derives an improvement of Schlickewei’s upper bound on the number of solutions of \(S\)-unit equations. The main tool of the proofs is an improvement of the quantitative subspace theorem given by the author [An improvement of the quantitative subspace theorem, Compos. Math. 101, No. 3, 225-311 (1996)].

Reviewer: I.Gaál (Debrecen)

##### Keywords:

\(S\)-unit equations; decomposable form equation; non-degenerate solutions; number of solutions; norm form##### References:

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