216 IV Continued Fractions and Their Uses
wherec 2 =(c 1 /|a 0 |)^1 /n.Ifk=j,then
|x−ζky|≥|(ζj−ζk)y|−|x−ζjy|
≥c 3 |y|−c 2 |y|m/n≥c 4 |y|,wherec 3 ,c 4 are positive constants. It follows that
|a 0 ||x−ζjy|cn 4 −^1 |y|n−^1 ≤|f(x,y)|=|g(x,y)|≤c 1 |y|mand hence
|ζj−x/y|≤c 5 /|y|n−m,where the positive constantc 5 depends only on the coefficients offandg. Evidently
this implies thatζjis real. Sinceζjis not rational andm≤n−3, we now obtain a
contradiction to Roth’s theorem.
It is actually possible to characterize all polynomial Diophantine equations with
infinitely many solutions. LetF(x,y)be a polynomial with rational coefficients which
is irreducible overC. It was shown by Siegel (1929), by combining his own results on
the approximation of algebraic numbers with results of Mordell and Weil concerning
the rational points on elliptic curves and Jacobian varieties, that if the equation
F(x,y)= 0 (∗)has infinitely many integer solutions, then there exist polynomials or Laurent poly-
nomialsφ(t),ψ(t)(not both constant) with coefficients from either the rational field
Qor a real quadratic fieldQ(
√
d),whered>0 is a square-free integer, such that
F(φ(t),ψ(t))is identically zero. Ifφ(t),ψ(t)are Laurent polynomials with coeffi-
cients fromQ(
√
d), they may be chosen to be invariant whentis replaced byt−^1 and
the coefficients are replaced by their conjugates inQ(
√
d).
This implies, in particular, that the algebraic curve defined by(∗)may be trans-
formed by a birational transformation with rational coefficients into either a linear
equationax+by+c=0 or a Pellian equationx^2 −dy^2 −m=0. It is not signif-
icant that the birational transformation has rational, rather than integral, coefficients
since, by combining a result of Mahler (1934) with theMordell conjecture,itmaybe
seen that the same conclusions hold if the equation(∗)has infinitely many solutions
in rational numbers whose denominators involve only finitely many primes.
The conjecture of Mordell (1922) says that the equation(∗)has at most finitely
manyrationalsolutions if the algebraic curve defined by(∗)has genusg>1. (The
concept ofgenuswill not be formally defined here, but we mention that the genus of an
irreducible plane algebraic curve may be calculated by a procedure due to M. Noether.)
The conjecture has now been proved by Faltings (1983), as will be mentioned in
Chapter XIII. As mentioned also at the end of Chapter XIII, if the algebraic curve
defined by(∗)has genus 1, then explicit bounds may be obtained for the number of
integral points. It was already shown by Hilbert and Hurwitz (1890) that the algebraic
curve defined by(∗)has genus 0 if and only if it is birationally equivalent overQ
either to a line or to a conic. There then exist rational functionsφ(t),ψ(t)(not both
constant) with coefficients either fromQor from a quadratic extension ofQsuch that