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Decomposable form equations with a small linear scattering. (English) Zbl 0754.11009

Let \(S=\{p_ 1,\ldots,p_ t\}\) be a finite, possibly empty, set of primes, \(\mathbb{Z}_ S=\mathbb{Z}[(p_ 1\dots p_ t)^{-1}]\) the ring of \(S\)-integers and \(\mathbb{Z}^*_ S\) the unit group of \(\mathbb{Z}_ S\). Further, let \(F({\mathfrak X})\in\mathbb{Z}_ S[X_ 1,\dots,X_ n]\) be a decomposable form, i.e. a homogeneous polynomial that can be factorized into linear forms with algebraic coefficients. We shall deal, among other things, with the decomposable form equation \[ F({\mathbf x})\in\mathbb{Z}^*_ S\quad\text{ in }{\mathbf x}\in\mathbb{Z}^ n_ S.\tag{1} \] The linear scattering of (1) is the smallest number \(h\) such that the set of solutions of (1) is contained in some union \(V_ 1\cup\cdots\cup V_ h\), where \(V_ 1,\ldots,V_ h\) are proper linear subspaces of \(\mathbb{Q}^ n\). We call two decomposable forms \(F\) and \(G\) \(S\)-equivalent if there is a linear transformation on the variables \((X_ 1,\ldots,X_ n)\), given by a matrix in \(GL_ n(\mathbb{Z}_ S)\), that transforms \(F\) into \(G\). The linear scattering of (1) is invariant under \(S\)-equivalence. The \(S\)- automorphism group \(\operatorname{Aut}(\mathbb{Z}^ n_ S,F)\) of a decomposable form \(F\) is defined as the group of all linear transformations on \((X_ 1,\dots,X_ n)\), given by a matrix in \(GL_ n(\mathbb{Z}_ S)\), that transform \(F\) into itself.
Among other things, we shall prove the following: for every \(r\geq 2\) and every algebraic number field \(L\), there are only finitely many \(S\)- equivalence classes of decomposable forms \(F\in\mathbb{Z}_ S[X_ 1,\dots,X_ n]\) of degree \(r\), that factor into linear forms in \(L[X_ 1,\dots,X_ n]\), that have finite \(S\)-automorphism group, and for which (1) has linear scattering \(\geq(n!)^{2n+2}\). This generalizes a result of K. Györy and the author [J. Reine Angew. Math. 399, 60-80 (1989; Zbl 0675.10009)] on Thue-Mahler equations.
The proof depends essentially on Schlickewei’s \(p\)-adic generalization of Schmidt’s Subspace Theorem; therefore, it does not allow one to determine effectively the collection of exceptional \(S\)-equivalence classes. We also consider (1) for decomposable forms \(F\) with infinite \(S\)- automorphism group, and show that in the theorem mentioned above, the condition that \(\operatorname{Aut}(\mathbb{Z}^ n_ S,F)\) be finite cannot be essentially relaxed. We generalize the results mentioned above to decomposable form equations over rings of \(S\)-integers of algebraic number fields.

MSC:

11D57 Multiplicative and norm form equations

Citations:

Zbl 0675.10009
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References:

[1] Stewart, Evertse Gy ry and Tijdeman On unit equations in two unknowns Invent, Math pp 92– (1988)
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