Thue-Mahler equation

Let $F ( X , Y ) \in \mathbf{Z} [ X , Y ]$ be a binary form of degree $r \geq 3$, irreducible over $\mathbf{Q}$, let $S = \{ p _ { 1 } , \dots , p _ { s } \}$ be a fixed set of rational prime numbers and let $a \neq 0$ be a fixed rational integer. The Diophantine equation (cf. also Diophantine equations)

\begin{equation} \tag{a1} F ( x , y ) = a p _ { 1 } ^ { z _ { 1 } } \dots p _ { s } ^ { z _ { s } } \end{equation}

in the unknowns $x , y , z _ { 1 } , \dots , z _ { s } \in \mathbf{Z}$, with $x$ and $y$ relatively prime, is called a Thue–Mahler equation. More generally, let $K$ be an algebraic number field (cf. Number field; Algebraic number), let $S$ be a fixed finite set of places in $K$ (cf. also Place of a field), containing all infinite ones, let $\mathcal{O} _ { S }$ be the ring of $S$-integers and let $\mathcal{O} _ { S } ^ { * }$ be the group of $S$-units of $K$. Let $F ( X , Y ) \in O _ { S } [ X , Y ]$ be a binary form of degree $r \geq 3$, irreducible over $K$. The Diophantine equation

\begin{equation} \tag{a2} F ( x , y ) \in \mathcal{O} _ { S } ^ { * }\quad \text { in} ( x , y ) \in \mathcal{O} _ { S } \times \mathcal{O} _ { S } \end{equation}

is called a generalized Thue–Mahler equation. If $a, p _ { 1 } , \dots , p _ { s }$ are as in (a1) and one takes in (a2) $K = \mathbf{Q}$ and $S = \{ p _ { 1 } , \dots , p _ { s } \} \cup \{ p : p \text{ is prime and divides } a\}$, then all solutions of (a1) are also solutions of (a2). Hence, any result concerning the solutions of (a1) applies also to those of (a2).

In 1933, K. Mahler, using his $p$-adic analogues of the methods of A. Thue [a7] and C.L. Siegel [a5], proved in [a3] that a Thue–Mahler equation (a1) has at most finitely many solutions. Because of the applied methods, this result is non-effective, i.e. it does not imply an explicit bound for either the size of the unknowns, or for the number of solutions. The development of Baker's theory (cf. also Gel'fond–Baker method) and its $p$-adic analogues made possible, in the 1970s, the proof of effective, though not explicit, bounds for the size of the unknowns; see [a4], Chap. 7. Subsequently, very explicit upper bounds for

\begin{equation*} \operatorname { max } \{ | x | , | y | , p _ { 1 } ^ { z _ { 1 } } \ldots p _ { s } ^ { z _ { s } } \} \end{equation*}

have been proved. A characteristic result of this type is due to Y. Bugeaud and K. Gőry [a1], in which the quantities $s$, $a$, , $r$, $H$, $h$, $R$ are involved; here, $H > 3$ is an upper bound for the absolute values of the coefficients of $F$ and $h$, $R$ are, respectively, the class number and the regulator of the number field generated (over $\mathbf{Q}$) by a root of the polynomial $F ( X , 1 )$ (cf. also Class field theory).

Due to techniques in Diophantine approximations, explicit upper bounds for the number of essentially distinct solutions have been proved for (a2), where two solutions $( x _ { 1 } , y _ { 1 } )$, $( x _ { 2 } , y _ { 2 } )$ are considered as essentially distinct if $( x _ { 2 } , y _ { 2 } )$ is not of the form $( \epsilon x _ { 1 } , \epsilon y _ { 1 } )$ for some $\epsilon \in \mathcal{O} _ { S } ^ { * }$. In view of the observation following (a2), such a bound is also valid for the number of solutions of (a1). Thus, Mahler's finiteness result has been considerably generalized and, what is more, in an explicit form. A characteristic result of this type is due to J.-H. Evertse [a2]: Let the cardinality of $S$ in (a2) be $s$. Then, the number of essentially distinct solutions $( x , y ) \in \mathcal{O} _ { S } \times \mathcal{O} _ { S }$ is at most $( 5 \times 10 ^ { 6 } r ) ^ { s }$.

In the early 1990s, constructive methods for the explicit computation of all solutions of a Thue–Mahler equation (a1) were developed by N. Tzanakis and B.M.M. de Weger [a8], [a9]. These are based on the theory (real and complex as well as $p$-adic) of linear forms in logarithms of algebraic numbers (cf. Linear form in logarithms) and reduction techniques, like the LLL-basis reduction algorithm and the computation of "small" vectors in a lattice (cf. also LLL basis reduction method). This method can, in principle, be extended to equations of the form (a2), as shown by N.P. Smart in [a6].

References

[a1]	Y. Bugeaud, K. Győry, "Bounds for the solutions of Thue–Mahler equations and norm form equations" Acta Arith. , 74 (1996) pp. 273–292
[a2]	J.-H. Evertse, "The number of solutions of the Thue–Mahler equation" J. Reine Angew. Math. , 482 (1997) pp. 121–149
[a3]	K. Mahler, "Zur Approximation algebraischer Zahlen, I: Ueber den grössten Primteiler binärer Formen" Math. Ann. , 107 (1933) pp. 691–730
[a4]	T.N. Shorey, R. Tijdeman, "Exponential Diophantine equations" , Tracts in Math. , 87 , Cambridge Univ. Press (1986)
[a5]	C.L. Siegel, "Approximation algebraischer Zahlen" Math. Z. , 10 (1921) pp. 173–213
[a6]	N.P. Smart, "Thue and Thue–Mahler equations over rings of integers" J. London Math. Soc. , 56 : 2 (1997) pp. 455–462
[a7]	A. Thue, "Ueber Annäherungswerte algebraischer Zahlen" J. Reine Angew. Math. , 135 (1909) pp. 284–305
[a8]	N. Tzanakis, B.M.M. de Weger, "Solving a specific Thue–Mahler equation" Math. Comp. , 57 (1991) pp. 799–815
[a9]	N. Tzanakis, B.M.M. de Weger, "How to explicitly solve a Thue–Mahler equation" Compositio Math. , 84 (1992) pp. 223–288