Baker's 1968 paper 'Contributions to the theory of diophantine equations I' founded the effective theory of the Thue equation F(x,y)=mF(x,y)=m for irreducible binary forms FF of degree n≥3n\ge 3: via his bounds for linear forms in logarithms it gave the first effectively computable bound max⁡(∣x∣,∣y∣)<C e(log⁡m)κ\max(|x|,|y|)<C\,e^{(\log m)^{\kappa}} (κ>n+1\kappa>n+1) for all solutions, and the first generally effective improvement on Liouville's 1844 rational-approximation theorem. The paper, and the note added in proof, left open the natural limits of this effective program: extending the effective method beyond Thue's equation to general Diophantine equations in two unknowns (an effective Siegel theorem), the optimal exponent and shape of the effective bounds, the best effective irrationality exponent for general algebraic numbers, and the classification of equations reducible to Thue's equation. Effective resolution of integral points is known for all affine curves of genus at most 1 and for many special higher-genus classes (superelliptic/cyclic covers, Bilu's étale-cover criterion, all but finitely many modular curves, many genus-2 families), but no algorithm is known for arbitrary two-variable equations, and a 2023 obstruction theorem limits the Baker–Bilu mechanism on general curves of genus at least 2; the optimal form of the effective bounds remains open and conjecturally equivalent to abc-type statements; the first power-saving improvement of the classical effective bound for Mordell's equation y2=x3+ky^2=x^3+k appeared in 2026. The effective irrationality exponent for general algebraic numbers of degree dd remains between the effective d−εd-\varepsilon (Baker–Feldman) and the ineffective 2+ε2+\varepsilon (Roth).

Background

A binary form is a homogeneous polynomial F(x,y)F(x,y) with integer coefficients. When FF is irreducible of degree n≥3n\ge 3, the equation F(x,y)=mF(x,y)=m, for a nonzero integer mm, is called a Thue equation. Thue proved in 1909 that such an equation has only finitely many integer solutions, by deriving it from a statement about rational approximation: an algebraic number α\alpha of degree n≥3n\ge 3 has only finitely many rational approximations p/qp/q with ∣α−p/q∣<q−μ|\alpha-p/q|<q^{-\mu} for a suitable exponent μ\mu. Liouville's elementary theorem already gives, effectively, the exponent μ=n\mu=n; Thue obtained a smaller exponent μ=n/2+1+ε\mu=n/2+1+\varepsilon, and Roth's theorem (1955) settled the optimal threshold μ=2+ε\mu=2+\varepsilon. All of these improvements of Liouville, however, were non-effective: they provide no computable constant cc with ∣α−p/q∣>c q−μ|\alpha-p/q|>c\,q^{-\mu}, and hence no computable bound on the solutions of F(x,y)=mF(x,y)=m. A result is called effective when its constants can, in principle, be computed from the input data.

Baker's 1968 paper made the theory effective. Building on his theory of linear forms in logarithms — explicit lower bounds for nonzero expressions b1log⁡α1+⋯+brlog⁡αrb_1\log\alpha_1+\dots+b_r\log\alpha_r with algebraic αi\alpha_i and rational integers bib_i — he proved (Theorem 1) that every integer solution of F(x,y)=mF(x,y)=m satisfies max⁡(∣x∣,∣y∣)<C e(log⁡m)κ\max(|x|,|y|)<C\,e^{(\log m)^{\kappa}} for any real κ>n+1\kappa>n+1, where CC is effectively computable in terms of nn, κ\kappa and the coefficients of FF; and (Theorem 2, formally deduced from Theorem 1) that for every algebraic α\alpha of degree n≥3n\ge 3 and any κ>n+1\kappa>n+1 one has, effectively, ∣α−p/q∣>c q−ne(log⁡q)1/κ|\alpha-p/q|>c\,q^{-n}e^{(\log q)^{1/\kappa}} — the first generally effective improvement on Liouville's theorem, albeit one that improves only the multiplicative factor by a sub-power of qq rather than the exponent. For special numbers stronger results were already known: Baker himself had proved in 1964, by the Thue–Siegel hypergeometric method — the construction of explicit polynomial approximations to binomial functions such as (1−x)1/3(1-x)^{1/3} from hypergeometric series — that ∣23−p/q∣>10−6q−2.955|\sqrt[3]{2}-p/q|>10^{-6}q^{-2.955}, an improvement of the exponent itself. The paper's number-theoretic lemmas also carried the condition κ>n+1\kappa>n+1, and an improvement of this condition and of the overall shape of the bound was left open.

Siegel's theorem (1929) generalizes finiteness far beyond Thue equations: an affine curve — equivalently, a polynomial equation in two unknowns — has only finitely many integer, and more generally SS-integral (denominators allowed to involve only primes from a fixed finite set SS), solutions whenever it has genus at least 1 with at least one point at infinity, or genus 0 with at least three points at infinity; here genus is the topological genus of the smooth projective model of the curve. Siegel's proof proceeds through the Mordell–Weil theorem — the theorem that the group of rational points on an abelian variety (for genus 1, the elliptic curve itself) is finitely generated — or through non-effective approximation, and yields no bounds. For rational points on curves of genus at least 2, the analogous finiteness statement is Faltings's theorem (formerly Mordell's conjecture), likewise non-effective in general. A recurring yardstick below is the abc conjecture: the assertion that for each ε>0\varepsilon>0 only finitely many coprime positive integer triples a+b=ca+b=c satisfy c>rad(abc)1+εc>\mathrm{rad}(abc)^{1+\varepsilon}, where rad(n)\mathrm{rad}(n) denotes the product of the distinct prime divisors of nn. Making the general theorem effective, i.e. extending Baker's effective method from Thue's equation to general equations F(x,y)=0F(x,y)=0 in two unknowns, was the central open direction the paper left, together with determining which equations reduce at all to Thue (or unit-type) equations, and with sharpening the effective approximation exponent toward the strength known for special numbers. These directions — algorithmic scope, reduction mechanisms, optimal bounds, and optimal effective approximation — together define Baker's effective program.

Problem Statement

Complete Baker's effective program for Diophantine equations in two unknowns. The single scientific objective is to replace the ineffective finiteness theorems for equations in two unknowns by effective procedures of best-possible quantitative strength. Its four linked components are:

(1) Scope of effectivity. Determine whether every polynomial Diophantine equation in two unknowns whose integer (SS-integral) solution set is finite — equivalently, after reduction to smooth geometrically integral components, every affine curve U=X−RU=X-R with negative Euler characteristic (that is, a smooth projective curve XX of genus gg with a finite set RR of points at infinity removed, satisfying χ(U)=2−2g−#R<0\chi(U)=2-2g-\#R<0, which by Siegel's theorem is exactly the condition for the set of SS-integral points to be finite apart from the degenerate case U≅GmU\cong\mathbb{G}_m), of any genus — admits an effective determination of all its integral points: an explicit, terminating procedure with an explicit height bound and effectively computable constants. The genus-at-most-1 cases are settled by Baker's method; the open core is genus at least 2 (and, for rational points, the effective form of Faltings's theorem known as the effective Mordell conjecture).

(2) Reduction mechanism. Characterize which affine curves (equivalently, which two-variable equations) are reducible to finitely many Thue equations, Thue-Mahler equations (Thue equations whose right-hand side has all its prime factors in a fixed finite set SS), or unit equations (linear equations such as a1u1+a2u2=1a_1u_1+a_2u_2=1 in unknowns uiu_i restricted to SS-units, i.e. elements whose prime factorization involves only primes from SS) — the mechanism through which Baker's method operates — and determine whether a general such reduction, or an entirely new effective mechanism, exists for curves of genus at least 2; in particular, delimit the reach of the Baker-Bilu étale-cover criterion (Bilu's 1995 sufficient condition: after base field extension the affine curve admits a connected finite étale cover — a finite covering that is locally an isomorphism — with a nondegenerate morphism to Gm×Gm\mathbb{G}_m\times\mathbb{G}_m) in view of the Landesman-Poonen obstruction (their 2023 theorem that every curve of genus at least 2 over C\mathbb{C} has a point xx such that no connected finite étale cover of X−{x}X-\{x\} admits a nonconstant morphism to the multiplicative group Gm\mathbb{G}_m).

(3) Optimal effective bounds. Determine the best possible form of the effective height bounds for the solutions of Thue and related equations — Thue-Mahler, Mordell (equations y2=x3+ky^2=x^3+k), superelliptic (equations yℓ=f(x)y^{\ell}=f(x) with deg⁡f≥3\deg f\ge 3), and norm-form equations (equations asserting that a norm, i.e. a product of Galois conjugates, of a linear form with integer coefficients in a number field equals a fixed nonzero integer); in particular, whether the dependence on the right-hand side, of the shape e(log⁡m)κe^{(\log m)^{\kappa}} in Baker's Theorem 1 (with any κ>n+1\kappa>n+1 as published; later sharpenings of the underlying linear-forms bounds improve the constants), can be improved to a polynomial-type dependence, as the abc conjecture predicts via the product-versus-sum folklore for S-unit equation bounds, and prove matching constructions or conditional equivalences.

(4) Optimal effective approximation exponent. Determine the best effective irrationality exponent for general algebraic numbers: for an algebraic number α\alpha of degree d≥3d\ge 3, let μeff(α)\mu_{\mathrm{eff}}(\alpha) denote the infimum of exponents μ\mu such that ∣α−p/q∣>c(α) q−μ|\alpha-p/q|>c(\alpha)\,q^{-\mu} holds for all integers p,qp,q with q>0q>0, where c(α)c(\alpha) is effectively computable (i.e. computable by a terminating procedure from the minimal polynomial of α\alpha). Determine the best achievable μeff(α)\mu_{\mathrm{eff}}(\alpha) for general algebraic α\alpha — known effectively only up to μeff(α)≤d−ε(α)\mu_{\mathrm{eff}}(\alpha)\le d-\varepsilon(\alpha) with an effective ε(α)>0\varepsilon(\alpha)>0 (Baker-Feldman), with Roth's ineffective optimum at 2+ε2+\varepsilon — i.e. effectivize Thue-Siegel-Roth strength for general algebraic numbers beyond special classes such as the binomial numbers a/br\sqrt[r]{a/b}, or prove that no effective exponent below the Baker-Feldman value d−ε(α)d-\varepsilon(\alpha) is possible in general.

Components (1) and (2) form the algorithmic strand; (3) and (4) are its two quantitative faces, linked by the duality through which Baker's Theorem 2 (approximation) is deduced from Theorem 1 (the Thue bound). Each of the four components is open at its stated level of generality; resolutions restricted to special classes of curves, special families of forms, or special classes of algebraic numbers do not settle the general case.

References

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