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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200801.png" /> be a [[Binary form|binary form]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200802.png" />, irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200803.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200804.png" /> be a fixed set of rational prime numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200805.png" /> be a fixed rational integer. The Diophantine equation (cf. also [[Diophantine equations|Diophantine equations]])
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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in the unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200807.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t1200809.png" /> relatively prime, is called a Thue–Mahler equation. More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008010.png" /> be an algebraic number field (cf. [[Number field|Number field]]; [[Algebraic number|Algebraic number]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008011.png" /> be a fixed finite set of places in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008012.png" /> (cf. also [[Place of a field|Place of a field]]), containing all infinite ones, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008013.png" /> be the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008014.png" />-integers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008015.png" /> be the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008016.png" />-units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008017.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008018.png" /> be a binary form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008019.png" />, irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008020.png" />. The Diophantine equation
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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]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a1} F ( x , y ) = a p _ { 1 } ^ { z _ { 1 } } \dots p _ { s } ^ { z _ { s } } \end{equation}
  
is called a generalized Thue–Mahler equation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008022.png" /> are as in (a1) and one takes in (a2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008024.png" />, 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 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
  
In 1933, K. Mahler, using his <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008025.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008026.png" />-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} \tag{a2} F ( x , y ) \in \mathcal{O} _ { S } ^ { * }\quad \text { in} ( x , y ) \in \mathcal{O} _ { S } \times \mathcal{O} _ { S } \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008027.png" /></td> </tr></table>
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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).
  
have been proved. A characteristic result of this type is due to Y. Bugeaud and K. Gőry [[#References|[a1]]], in which the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008034.png" /> are involved; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008035.png" /> is an upper bound for the absolute values of the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008038.png" /> are, respectively, the class number and the regulator of the number field generated (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008039.png" />) by a root of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008040.png" /> (cf. also [[Class field theory|Class field theory]]).
+
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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008042.png" /> are considered as essentially distinct if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008043.png" /> is not of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008044.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008045.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008046.png" /> in (a2) be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008047.png" />. Then, the number of essentially distinct solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008048.png" /> is at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008049.png" />.
+
\begin{equation*} \operatorname { max } \{ | x | , | y | , p _ { 1 } ^ { z _ { 1 } } \ldots p _ { s } ^ { z _ { s } } \} \end{equation*}
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120080/t12008050.png" />-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]]].
+
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====
 
====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>
+
<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>

Latest revision as of 17:46, 1 July 2020

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
How to Cite This Entry:
Thue-Mahler equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thue-Mahler_equation&oldid=23081
This article was adapted from an original article by N. Tzanakis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article