View source for Thue-Mahler equation

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 50 formulas, 49 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|part}}
Let $F ( X , Y ) \in \mathbf{Z} [ X , Y ]$ be a [[Binary form|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|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|Number field]]; [[Algebraic number|Algebraic number]]), let $S$ be a fixed finite set of places in $K$ (cf. also [[Place of a field|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 [[#References|[a7]]] and C.L. Siegel [[#References|[a5]]], proved in [[#References|[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|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 [[#References|[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 [[#References|[a1]]], in which the quantities $s$, $a$, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008030.png"/>, $r$, $H$, $h$, $R$ are involved; here, $H &gt; 3$ 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|Class field theory]]).

Due to techniques in [[Diophantine approximations|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 [[#References|[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 [[#References|[a8]]], [[#References|[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|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|LLL basis reduction method]]). This method can, in principle, be extended to equations of the form (a2), as shown by N.P. Smart in [[#References|[a6]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  Y. Bugeaud,   K. Győry,   "Bounds for the solutions of Thue–Mahler equations and norm form equations"  ''Acta Arith.'' , '''74'''  (1996)  pp. 273–292</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.-H. Evertse,   "The number of solutions of the Thue–Mahler equation"  ''J. Reine Angew. Math.'' , '''482'''  (1997)  pp. 121–149</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  K. Mahler,   "Zur Approximation algebraischer Zahlen, I: Ueber den grössten Primteiler binärer Formen"  ''Math. Ann.'' , '''107'''  (1933)  pp. 691–730</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T.N. Shorey,   R. Tijdeman,   "Exponential Diophantine equations" , ''Tracts in Math.'' , '''87''' , Cambridge Univ. Press  (1986)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C.L. Siegel,   "Approximation algebraischer Zahlen"  ''Math. Z.'' , '''10'''  (1921)  pp. 173–213</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  N.P. Smart,   "Thue and Thue–Mahler equations over rings of integers"  ''J. London Math. Soc.'' , '''56''' :  2  (1997)  pp. 455–462</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A. Thue,   "Ueber Annäherungswerte algebraischer Zahlen"  ''J. Reine Angew. Math.'' , '''135'''  (1909)  pp. 284–305</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  N. Tzanakis,   B.M.M. de Weger,   "Solving a specific Thue–Mahler equation"  ''Math. Comp.'' , '''57'''  (1991)  pp. 799–815</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  N. Tzanakis,   B.M.M. de Weger,   "How to explicitly solve a Thue–Mahler equation"  ''Compositio Math.'' , '''84'''  (1992)  pp. 223–288</td></tr></table>