Siegel method

A method for investigating the arithmetical properties of the values assumed at algebraic points by $E$-functions that satisfy linear differential equations with coefficients in ; first proposed by C.L. Siegel [1].

An entire function

is called an -function if all its coefficients belong to an algebraic field of finite degree (cf. Algebraic number; Field), if for every the maximum modulus of the is and if there exists a sequence of rational integers such that is an algebraic integer for . Examples are , and the Bessel function (cf. Bessel functions).

Let , and , . If and are rational numbers, and , then the function

is an -function; it satisfies a linear differential equation of order with coefficients in .

Siegel's main result pertains to the values of the function

where is the Bessel function. If is a rational number, then for any algebraic number the numbers and are algebraically independent over (cf. Algebraic independence).

In 1949 Siegel presented his method in a general setting, but the conditions that had to be imposed on the -functions so that their values could be assumed algebraically independent proved to be very hard to check. He was therefore unable to achieve any concrete new results.

Further development and generalization of Siegel's method should be credited to A.B. Shidlovskii (see [2]–[3]): Let be -functions which constitute a solution to the system of differential equations

(1)

and let be an algebraic number distinct from zero and from the singular points of the system (1); then the numbers are algebraically independent over if and only if the functions are algebraically independent over . This theorem implies, in particular, that if are algebraically independent, then all the numbers are transcendental (cf. Transcendental number); the same holds for all non-zero -points of the functions distinct from the poles of the system (1), provided is algebraic. The theorem has produced a great number of results concerning specific -functions, and algebraic independence proofs for values of -functions satisfying linear homogeneous and inhomogeneous differential equations of order higher than two. For example, the function

satisfies a linear differential equation of order with coefficients in ; it can be proved that for any algebraic number , the numbers , ; , are algebraically independent over .

Under the same conditions, the maximum number of numbers which are algebraically independent over is equal to the maximum number of functions which are algebraically independent over . If are -functions that are algebraically independent over and that satisfy the system (1), then for all points , with the possible exception of finitely many, the numbers are algebraically independent over . In each specific case the exceptional points can actually be determined.

These theorems provide the solution to virtually all problems of a general nature concerning transcendence and algebraic independence of the values of -functions at algebraic points.

Siegel's methods enables one to estimate the measure of algebraic independence of the numbers , thus giving the results a quantitative form. If the functions are algebraically independent, then , where is independent of and depends only on and the degree of the algebraic number .

References

[1]	C.L. Siegel, "Ueber einige Anwendungen Diophantischer Approximationen" Abh. Deutsch. Akad. Wiss. Phys.- Math. Kl. : 1 (1929) pp. 1–41
[2]	A.B. Shidlovskii, "On tests for algebraic independence of the values of a class of entire functions" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 1 (1959) pp. 35–66 (In Russian)
[3]	A.B. Shidlovskii, "On transcendency and algebraic independence of values of -functions related with an arbitrary number of algebraic equations in the rational function field" Izv. Akad. Nauk SSSR Ser. Mat. , 26 : 6 (1962) pp. 877–910 (In Russian)
[4]	A.B. Shidlovskii, "On arithmetic properties of values of analytic functions" Proc. Steklov Inst. Math. , 132 (1972) pp. 193–233 Trudy Mat. Inst. Steklov. , 132 (1972) pp. 169–202
[5]	S. Lang, "A transcendence measure for -functions" Mathem. , 9 (1962) pp. 157–161
[6]	N.I. Fel'dman, A.B. Shidlovskii, "The development and present state of the theory of transcendental numbers" Russian Math. Surveys , 22 : 3 (1967) pp. 1–79 Uspekhi Mat. Nauk , 22 : 3 (1967) pp. 3–81

Comments

In the article above, is the measure of algebraic independence, cf. Algebraic independence, measure of.

References

[a1]	A.B. Shidlovskii, "Transcendental numbers" , de Gruyter (1989) (Translated from Russian)
[a2]	Y. André, "-functions" , Vieweg (1989)