Gel'fond-Baker method

for Diophantine equations

One of the most efficient methods for obtaining explicit upper bounds on the size of integer solutions of some broad classes of Diophantine equations arises from the theory of transcendental numbers, from the solution by A.O. Gel'fond of Hilbert's seventh problem and from subsequent work of A. Baker. The Gel'fond–Baker method also incorporates ideas due to C.L. Siegel, K. Mahler and S. Lang. From this method, explicit upper bounds have been derived for:

1) integer points on curves of genus $ 0 $ and $ 1 $( effective versions of Siegel's theorem, due to Baker, J. Coates and W.M. Schmidt); [a2], Chapt. IV, Sect. 12;

2) Thue equations $ F ( x,y ) = k $, where $ F $ is a homogeneous polynomial with algebraic coefficients, $ k $ is a fixed non-zero algebraic number, and the polynomial $ F ( X,1 ) $ has at least three distinct roots; [a1], Chapt. 5; [a2], Chapt. 3; [a3], Chapt. 4–5;

3) hyper-elliptic and super-elliptic equations, namely $ y ^ {k} = f ( x ) $ where $ k $ is a fixed integer and $ f ( x ) $ is a polynomial with algebraic coefficients, which define curves of positive genus. This is the case when $ k = 2 $ and $ f ( x ) $ has at least three distinct roots or when $ k \geq 3 $ and $ f ( x ) $ has at least two distinct roots; [a2], Chapt. 4; [a3], Chapt. 6. The method also applies to some classes of norm-form equations in more than two variables (by the work of K. Győry).

The bounds obtained from this method are typically extremely large, and some further results on Diophantine approximations are required to be able to completely determine all the solutions. Upper estimates are obtained for the height of the solutions, either in the ring of integers of a fixed algebraic number field $ K $, or, more generally, in the ring of $ S $- integers, where $ S $ is a fixed finite set of prime ideals of $ K $( cf. also Ideal; Prime ideal). This means that a denominator for $ x $ and $ y $ is allowed, but is required to be the product of primes from a fixed finite set $ S $. This generalizes ordinary integers and units, which arise from taking for $ S $ the empty set. For instance, for the field of rational numbers, given a finite set $ S = \{ p _ {1} \dots p _ {s} \} $ of prime numbers, the $ S $- integers are the rational numbers $ {a / b } $ such that all prime divisors of the denominator $ b $ belong to $ S $, while the $ S $- units are the rational numbers of the form $ p _ {1} ^ {k _ {1} } \dots p _ {s} ^ {k _ {s} } $ with $ k _ {j} \in \mathbf Z $( $ 1 \leq j \leq s $).

Furthermore, the method can be extended in some cases to allow the exponents on $ x $ and $ y $ to vary. The most celebrated example is Tijdeman's theorem on the Catalan equation [a1], Chapt. 12, [a3], Chapt. 7, Sect. 3: There are only finitely many tuples $ ( x,y,p,q ) $ of integers, all of them $ \geq 2 $, satisfying $ x ^ {p} - y ^ {q} = 1 $.

According to Pillai's conjecture [a1], p. 201, [a2], p. 207, for each $ k \geq 2 $ the same should hold for the equation $ x ^ {p} - y ^ {q} = k $.

To demonstrate Baker's method, consider the Thue equation $ F ( x,y ) = k $, where $ k $ is a fixed non-zero algebraic number, while $ F $ is a homogeneous polynomial with algebraic coefficients such that $ F ( X,1 ) $ has at least three distinct roots, $ \alpha _ {1} $, $ \alpha _ {2} $, $ \alpha _ {3} $. The unknowns $ x $ and $ y $ are algebraic integers in a number field $ K $, which is assumed to contain not only the number $ k $, but also the coefficients of $ F $ as well as the roots of $ F ( X,1 ) $. Then one may write, for $ i = 1,2,3 $, $ x - \alpha _ {i} y = k _ {i} u _ {i} $, where $ k _ {i} $ belongs to a fixed finite subset of $ K $( independent of the solution $ ( x,y ) $), while $ u _ {1} $, $ u _ {2} $, $ u _ {3} $ are (unknown) units in $ K $. Eliminating $ x $ and $ y $ from these three equations yields the relation

$$ k _ {1} ( \alpha _ {3} - \alpha _ {2} ) u _ {1} + k _ {2} ( \alpha _ {1} - \alpha _ {3} ) u _ {2} + k _ {3} ( \alpha _ {2} - \alpha _ {1} ) u _ {3} = 0. $$

Therefore, a key step in the Gel'fond–Baker method is the following result, which deals with the $ S $- unit equation: If $ K $ is a number field and $ S $ is a finite set of prime ideals of $ K $, the equation $ x + y = 1 $ has only finitely many solutions in $ S $- units $ ( x,y ) $ of $ K $. Moreover, explicit bounds for the size of the solutions can be given.

The proof of this result relies on an estimate from transcendental number theory. Write $ x $ and $ y $ using a basis $ \epsilon _ {1} \dots \epsilon _ {r} $ of the group of $ S $- units modulo torsion. Then a "large" solution $ ( x,y ) $ to the Diophantine equation $ F ( x,y ) = k $ gives rise to a "small" value for a number of the form $ | {\zeta \epsilon _ {1} ^ {b _ {1} } \dots \epsilon _ {r} ^ {b _ {r} } - 1 } | $, where $ \zeta $ is a root of unity and $ b _ {1} \dots b _ {r} $ are rational integers. The so-called "theory of linear forms in logarithms" (cf. Linear form in logarithms) provides a lower bound for such numbers, which is sharp in terms of $ \max \{ | {b _ {1} } | \dots | {b _ {r} } | \} $. This lower bound, together with the link between large solutions and small values, then provides an upper bound for the size of solutions.

The Schmidt subspace theorem and its $ p $- adic variants by H.P. Schlickewei [a2], Chapt.5, Sect. 1, imply more generally that for each $ n \geq 2 $, the Diophantine equation $ x _ {1} + \dots + x _ {n} = 1 $ has only finitely many solutions in $ S $- units of $ K $( one considers only solutions where no subsum on the left-hand side vanishes). But for $ n \geq 3 $ the result is not effective: upper bounds for the number of solutions can be given, but no bound for the size of the solutions is known in general.

References

[a1]	T.N. Shorey, R. Tijdeman, "Exponential diophantine equations" , Tracts in Math. , 87 , Cambridge Univ. Press (1986)
[a2]	W.M. Schmidt, "Diophantine approximations and Diophantine equations" , Lecture Notes in Mathematics , 1467 , Springer (1991)
[a3]	V.G. Sprindžuk, "Classical diophantine equations" , Lecture Notes in Mathematics , 1559 , Springer (1993) (In Russian)