Diophantine approximation, problems of effective

2020 Mathematics Subject Classification: Primary: 11J [MSN][ZBL]

Ways of obtaining effective solutions of problems in Diophantine approximations for which a solution obtained by non-effective methods — i.e. by methods by which the result cannot be numerically expressed — is known. They include, for example, the theorems of A. Thue, C.L. Siegel, K. Roth, W. Schmidt, their generalizations, analogues and corollaries (cf. Thue–Siegel–Roth theorem; Diophantine approximations). The non-effectiveness of these theorems is due to the logical structure of a method based on the assumed existence of objects which are not constructively defined. Thus, in the case of rational approximations to algebraic numbers, the bound for the denominators of "good" approximations, which is established as a result of the reasoning, depends on one of these "good" approximations, the existence of which is not proved.

An effective solution of the problem often involves major difficulties. An effective strengthening of Liouville's inequality (cf. Liouville number) could be obtained only recently. The method of proof strongly differs from the methods of Thue–Siegel–Roth and involves the application of effective methods of the theory of transcendental numbers (cf. Linear form in logarithms). The best result thus far (1978) known has the form

$$|\alpha x-y|>cx^{-n+1+\delta},$$

where $\alpha$ is an algebraic number of degree $n\geq3$, $x>0$, $y$ are rational integers, and $c>0$ and $\delta>0$ are explicitly defined in terms of $\alpha$ [3]. This inequality is very different from its non-effective analogue: Instead of the exponent $-n+1+\delta$, non-effective methods yield $-1-\epsilon$ with an arbitrary $\epsilon>0$, but with an unknown function $c$ of $\alpha$ and $\epsilon$. The proof of the effective inequality

$$|\alpha x-y|>Cx^{-\phi(n)}$$

with an increasing function $\phi(n)$, for example increasing like $n^\epsilon$, is of major interest because of the relation with bounds on the solutions of a Diophantine equation

$$f(x,y)=0,$$

where the polynomial $f(x,y)$ defines a curve of genus $\geq1$ (Siegel in 1929 proved that the number of solutions was finite by using non-effective estimates; cf. Diophantine geometry).

Even though the quality of the effective estimates is much worse than that of the ineffective, knowledge of the dependence of the former on the parameters of the problem yields new results which are unobtainable with the latter. Thus, effective estimates of linear forms in logarithms of algebraic numbers allowed A. Baker to find estimates of solutions of many Diophantine equations, in particular of Thue's equation and of equations specifying curves of genus one, and also to give another solution of the problem of the tenth discriminant; to establish a bound for the discriminants of imaginary quadratic fields of class 2; to estimate from below the largest prime divisor of the values of a binary form of degree $\geq3$ and the value of the square-free kernel of an integer polynomial [2].

References

[1a]	V.G. Sprindzhuk, "Rational approximations to algebraic numbers" Math. USSR Izv. , 5 : 5 (1971) pp. 1003–1020 Izv. Akad. Nauk. SSSR Ser. Mat. , 35 : 5 (1971) pp. 991–1007
[1b]	V.G. Sprindzhuk, "On an estimate for solutions of Thue's equation" Math. USSR Izv. , 6 : 4 (1972) pp. 705–734 Izv. Akad. Nauk. SSSR Ser. Mat. , 36 : 4 (1972) pp. 712–741
[2]	V.G. Sprindzhuk, "The metric theory of Diophantine approximations" , Current problems of analytic number theory , Minsk (1974) pp. 178–198 (In Russian)
[3]	N.I. Fel'dman, "Estimates of linear forms in logarithms of algebraic numbers, and some applications of them" , Current problems of analytic number theory , Minsk (1974) pp. 244–268 (In Russian)
[4]	N.I. Fel'dman, "An effective refinement of the exponent in Liouville's theorem" Math. USSR Izv. , 5 : 5 (1971) pp. 985–1002 Izv. Akad. Nauk. SSSR Ser. Mat. , 35 : 5 (1971) pp. 937–990
[5a]	A. Baker, "Contributions to the theory of Diophantine equations I: on the representation of integers by binary forms" Phil. Trans. Royal Soc. London Ser. A , 263 (1968) pp. 173–191
[5b]	A. Baker, "Contributions to the theory of Diophantine equations II: The Diophantine equation $y^2=x^3+k$" Phil. Trans. Royal Soc. London Ser. A , 263 (1968) pp. 193–208
[6]	A. Baker, "Effective methods in the theory of numbers" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 19–26

Comments

The number $\delta$ obtained by Baker's method is usually extremely small. In special cases one can improve its value considerably by other methods; see, for example, [a1]–[a3].

The problem of the 10th discriminant refers to the existence or non-existence of a tenth imaginary quadratic field $\mathbf Q(\sqrt{-d})$ of class number one besides the nine known ones. It turned out not to exist, cf. Quadratic field.

References

[a1]	A. Baker, "Rational approximations to $2^{1/3}$ and other algebraic numbers" Quart. J. Math. Oxford , 15 (1964) pp. 375–383
[a2]	E. Bombieri, "On the Thue–Siegel–Dyson theorem" Acta. Math. , 148 (1982) pp. 255–296
[a3]	G. Chudnovsky, "On the method of Thue–Siegel" Ann. of Math. , 117 (1983) pp. 325–382
[a4]	A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975)