Gel'fond-Schneider method

In 1934 Hilbert's seventh problem (cf. also Hilbert problems) was solved independently by A.O. Gel'fond [a3] and Th. Schneider [a8]: If $\alpha$ is a non-zero algebraic number, $\operatorname { log } \alpha$ a non-zero logarithm of $\alpha$ and $\beta$ an irrational algebraic number, then the number $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ is transcendental (cf. Transcendental number).

The transcendence of $e ^ { \pi }$ (corresponding to $\alpha = - 1$, $\operatorname { log } \alpha = i \pi$, $\beta = - i$) had already been proved by Gel'fond in 1929 [a2] using interpolation formulas for the function $e ^ { \pi z }$, like in Pólya's work [a7] on integral-valued entire functions.

One main common feature of both the Gel'fond and the Schneider method is to start with the construction of an auxiliary function by means of Dirichlet's box principle (the Thue–Siegel lemma; cf. also Dirichlet principle).

While Schneider's proof (cf. Schneider method) is based on the addition theorem for the exponential function $e ^ { z _ { 1 } + z _ { 2 } } = e ^ { z _ { 1 } } e ^ { z _ { 2 } }$, the main ingredient in Gel'fond's proof is the differential equation $( d / d z ) e ^ { z } = e ^ { z }$. Gel'fond considers the two functions $e ^ { z }$ and $e ^ { \beta z }$; his auxiliary function has the form $F ( z ) = P ( e ^ { z } , e ^ { \beta z } )$, where $P$ is a polynomial with algebraic coefficients. He investigates the values of $F$ as well as its derivatives at the points $s\operatorname{log} \alpha$, $s \in \mathbf{Z}$. An extrapolation is an essential feature of his proof.

This method has been developed by Gel'fond himself for proving quantitative Diophantine approximation estimates (see [a4]; see also Gel'fond–Baker method; Diophantine approximations), and by Schneider, who obtained an extension of the Gel'fond–Schneider theorem to elliptic and Abelian functions: he proved the transcendence of elliptic integrals of the first or second kind [a9] and of Abelian integrals [a10], including the transcendence of the values $B ( a , b )$ of the beta-function at rational points $( a , b ) \in ( \mathbf{Q} \backslash \mathbf{Z} ) ^ { 2 }$. Next, Schneider [a11], [a12] provided general statements on the algebraic values of analytic functions satisfying differential equations; these results have been simplified and improved in the 1960s by S. Lang [a5], who extended Schneider's results to commutative algebraic groups. The following far-reaching statement is called the Schneider–Lang criterion: Let $K$ be a number field and let $f _ { 1 } , \ldots , f _ { d }$ be meromorphic functions in $\mathbf{C}$ of finite order of growth (cf. also Meromorphic function). Assume $f _ { 1 }$, $f _ { 2 }$ are algebraically independent (cf. also Algebraic independence). Assume also that for $i = 1 , \ldots , d$, the derivative $( d / d z ) f _ { i }$ of $f_i$ belongs to the ring $K [ f _ { 1 } , \ldots , f _ { d } ]$. Then the set of $w \in \mathbf{C}$ that are not poles of any $f _ { 1 } , \ldots , f _ { d }$ and such that $f _ { i } ( w ) \in K$, for $1 \leq i \leq d$, is finite.

Schneider and Lang extended their criterion to several variables by considering Cartesian products; a deeper result, involving algebraic hypersurfaces and suggested by M. Nagata [a5], has been obtained by E. Bombieri [a1].

A clever modification of the Gel'fond–Schneider method has been applied to modular functions in [a13], solving Mahler's conjecture: For any algebraic number $\alpha$ with $0 < | \alpha | < 1$ the value $J ( \alpha )$ of the modular function is transcendental.

Gel'fond proved in 1949 the algebraic independence of $2 ^{\sqrt [ 2 ] { 3 }}$ and $2^{ \sqrt [ 4 ] { 3 }}$. More generally, he proved that for algebraic $\alpha$ and $\beta$ with $\alpha \notin \{ 0,1 \}$ and $\beta$ of degree $d \geq 3$, the transcendence degree over $\mathbf{Q}$ of the field $\mathbf{Q} ( \alpha ^ { \beta } , \ldots , \alpha ^ { \beta ^ { d - 1 } } )$ is $\geq 2$ (cf. also Transcendental extension). After the work of G.V. Chudnovskii, P. Philippon and G. Diaz, it is known that this transcendence degree is $\geq [ ( d + 1 ) / 2 ]$. This method not only provides a new proof of the Lindemann–Weierstrass theorem on the algebraic independence of numbers $e ^ { \beta _ { 1 } } , \ldots , e ^ { \beta _ { n } }$ when $\beta _ { 1 } , \ldots , \beta _ { n }$ are $\mathbf{Q}$-linearly independent algebraic numbers, but also yields a similar result for elliptic functions (and, more generally, Abelian functions), as shown by Philippon and G. Wüstholz.

Also, Chudnovskii proved the algebraic independence of the two numbers $\pi$, $\Gamma ( 1 / 4 )$ (showing therefore that $\Gamma ( 1 / 4 )$ is transcendental), and later Yu.V. Nesterenko adapted the method of [a13] and obtained remarkable results of algebraic independence on values of modular functions, including the algebraic independence of the three numbers $\pi$, $\Gamma ( 1 / 4 )$ and $e ^ { \pi }$ [a6].

In another direction, both the Gel'fond and the Schneider method have been extended in order to prove results of linear independence over the field of algebraic numbers of logarithms of algebraic numbers (see Schneider method and Gel'fond–Baker method).

References

[a1]	E. Bombieri, "Algebraic values of meromorphic maps" Invent. Math. , 10 (1970) pp. 267–287 (Addendum, 11 (1970), 163-166)
[a2]	A.O. Gel'fond, "Sur les propriétés arithmétiques des fonctions entières" Tôhoku Math. J. , 30 (1929) pp. 280–285
[a3]	A.O. Gel'fond, "Sur le septième problème de Hilbert" Izv. Akad. Nauk. SSSR , 7 (1934) pp. 623–630 Dokl. Akad. Nauk. SSSR , 2 (1934) pp. 1–6
[a4]	A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover (1960) (In Russian)
[a5]	S. Lang, "Introduction to transcendental numbers" , Addison-Wesley and Don Mills (1966) (reprinted in: Collected Papers, Vol. I, Springer, 2000, pp. 396-506) MR0214547 Zbl 0144.04101
[a6]	"Introduction to algebraic independence theory. Instructional Conference (CIRM Luminy, 1997)" Y.V. Nesterenko (ed.) P. Philippon (ed.) , Lecture Notes in Mathematics , 1752 , Springer (2001)
[a7]	G. Pólya, "Über ganzwertige ganze Funktionen" Rend. Circ. Mat. Palermo , 40 (1915) pp. 1–16 (See also: Collected papers I Singularities of analytic functions, (ed. R.P. Boas), MIT (1974), 1-16)
[a8]	Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen I" J. Reine Angew. Math. , 172 (1934) pp. 65–69 Zbl 0010.10501 Zbl 60.0163.03
[a9]	Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen II" J. Reine Angew. Math. , 172 (1934) pp. 70–74
[a10]	Th. Schneider, "Zur Theorie der Abelschen Funktionen und Integrale" J. Reine Angew. Math. , 183 (1941) pp. 110–128
[a11]	Th. Schneider, "Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise" Math. Ann. , 121 (1949) pp. 131–140
[a12]	Th. Schneider, "Einführung in die transzendenten Zahlen" , Springer (1957)
[a13]	K. Barré–Sirieix, G. Diaz, F. Gramain, G. Philibert, "Une preuve de la conjecture de Mahler–Manin" Invent. Math. , 124 : 1–3 (1996) pp. 1–9