In mathematics, a Thue equation is a Diophantine equation of the form
is an irreducible bivariate form of degree at least 3 over the rational numbers, and
is a nonzero rational number.
It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers
, a result known as Thue's theorem.
[1] The Thue equation is solvable effectively: there is an explicit bound on the solutions
depend only on the form
A stronger result holds: if
is the field generated by the roots of
, then the equation has only finitely many solutions with
[2] Thue's original proof that the equation named in his honour has finitely many solutions is through the proof of what is now known as Thue's theorem: it asserts that for any algebraic number
there exists only finitely many coprime integers
Applying this theorem allows one to almost immediately deduce the finiteness of solutions.
However, Thue's proof, as well as subsequent improvements by Siegel, Dyson, and Roth were all ineffective.
Finding all solutions to a Thue equation can be achieved by a practical algorithm,[3] which has been implemented in the following computer algebra systems: While there are several effective methods to solve Thue equations (including using Baker's method and Skolem's p-adic method), these are not able to give the best theoretical bounds on the number of solutions.
One may qualify an effective bound
The best result known today, essentially building on pioneering work of Bombieri and Schmidt,[4] gives a bound of the shape
is an absolute constant (that is, independent of both
is the number of distinct prime factors of
The most significant qualitative improvement to the theorem of Bombieri and Schmidt is due to Stewart,[5] who obtained a bound of the form
but not its coefficients, and completely independent of the integer
on the right hand side of the equation.
This is a weaker form of a conjecture of Stewart, and is a special case of the uniform boundedness conjecture for rational points.
This conjecture has been proven for "small" integers
, where smallness is measured in terms of the discriminant of the form
, by various authors, including Evertse, Stewart, and Akhtari.
Stewart and Xiao demonstrated a strong form of this conjecture, asserting that the number of solutions is absolutely bounded, holds on average (as
ranges over the interval
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