Thue equation

In mathematics, a Thue equation is a Diophantine equation of the form

$$ f(x,y) = r, $$

where $$f$$ is an irreducible bivariate form of degree at least 3 over the rational numbers, and $$r$$ 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 $$x$$ and $$y$$, a result known as Thue's theorem,

The Thue equation is solvable effectively: there is an explicit bound on the solutions $$x$$, $$y$$ of the form $$(C_1 r)^{C_2}$$ where constants $$C_1$$ and $$C_2$$ depend only on the form $$f$$. A stronger result holds: if $$K$$ is the field generated by the roots of $$f$$, then the equation has only finitely many solutions with $$x$$ and $$y$$ integers of $$K$$, and again these may be effectively determined.

Finiteness of solutions and diophantine approximation
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 $$\alpha$$ having degree $$d \geq 3$$ and for any $$\varepsilon > 0 $$ there exists only finitely many co-prime integers $$p, q$$ with $$ q > 0 $$ such that $$ |\alpha - p/q| < q^{-(d+1+\varepsilon)/2}$$. 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.

Solution algorithm
Finding all solutions to a Thue equation can be achieved by a practical algorithm, which has been implemented in the following computer algebra systems:
 * in PARI/GP as functions thueinit and thue.
 * in Magma as functions ThueObject and ThueSolve.
 * in Mathematica through Reduce[]
 * in Maple through ThueSolve

Bounding the number of solutions
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 $$C(f,r)$$ of the Thue equation $$f(x,y) = r$$ by the parameters it depends on, and how "good" the dependence is.

The best result known today, essentially building on pioneering work of Bombieri and Schmidt, gives a bound of the shape $$C(f,r) = C \cdot (\deg f)^{1 + \omega(r)}$$, where $$C$$ is an absolute constant (that is, independent of both $$f$$ and $$r$$) and $$\omega(\cdot)$$ is the number of distinct prime divisors of $$r$$. The most significant qualitative improvement to the theorem of Bombieri and Schmidt is due to Stewart, who obtained a bound of the form $$C(f,r) = C \cdot (\deg f)^{1 + \omega(g)}$$ where $$g$$ is a divisor of $$r$$ exceeding $$|r|^{3/4}$$ in absolute value. It is conjectured that one may take the bound $$C(f,r) = C(\deg f)$$; that is, depending only on the degree of $$f$$ but not its coefficients, and completely independent of the integer $$r$$ 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 $$r$$, where smallness is measured in terms of the discriminant of the form $$f$$, 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 $$r$$ ranges over the interval $$|r| \leq Z $$ with $$Z \rightarrow \infty$$)