Schneider method

The name "Schneider method" arises from the solution, by Th. Schneider [a5] in 1934, to Hilbert's seventh problem (cf. also Hilbert problems): If $\alpha$ is a non-zero algebraic number, $\log \alpha$ a non-zero logarithm of $\alpha$ and $\beta$ an irrational algebraic number, then $\alpha^\beta$ is a transcendental number.

One main idea in Schneider's proof is to investigate values of a function $F(z) =P(z,\alpha^z)$ at points $u + v\beta$ ($u,v \in \mathbb{Z}$), where $P$ is a polynomial with algebraic coefficients. Assuming $\alpha^\beta$ is algebraic, a non-zero polynomial $P$ is constructed so that $F$ vanishes at many such points $u + v\beta$; this construction is based on Dirichlet's box principle or pigeon hole principle (the Thue–Siegel lemma, cf. also Transcendency, measure of; Dirichlet principle). A clever computation of a determinant is an essential tool of Schneider's proof.

A slight modification of the same argument yields the so-called six exponentials theorem [a2], [a4]: If $x_1,x_2$ are two complex numbers that are linearly independent over $\mathbb{Q}$ (cf. also Linear independence) and if $y_1,y_2,y_3$ are three complex numbers that are linearly independent over $\mathbb{Q}$ , then at least one of the six numbers $$ \exp(x_1y_1),\,\exp(x_1y_2),\,\exp(x_1y_3),\,\exp(x_2y_1),\,\exp(x_2y_2),\,\exp(x_2y_3) $$ is transcendental.

In relation with this statement, one of the simplest open (as of 2000) problems which would follow from the conjecture that "linearly independent logarithms of algebraic numbers are algebraically independent" is the four exponentials conjecture: If $x_1,x_2$ are two complex numbers that are linearly independent over $\mathbb{Q}$ and if $y_1,y_2$ are also two complex numbers that are linearly independent over $\mathbb{Q}$, then at least one of the four numbers $$ \exp(x_1y_1),\,\exp(x_1y_2),\,\exp(x_2y_1),\,\exp(x_2y_2) $$ is transcendental.

Both S. Lang [a2] and K. Ramachandra [a4] have given general statements concerning the simultaneous algebraic values of analytic functions by means of Schneider's method. Typically, these statements are efficient for functions satisfying functional equations, for instance when $f(z_1)$, $f(z_2)$ and $f(z_1+z_2)$ are algebraically dependent. This method yields transcendence results as well as results of algebraic independence on values of exponential, elliptic or Abelian functions; more generally it applies to the arithmetic study of commutative algebraic groups.

Schneider's method can be extended to several variables, and then yields partial results related to the Leopoldt conjecture on the $p$-adic rank of the units of an algebraic number field [a6]. It also gives lower bounds for the ranks of matrices whose entries are linear combinations with algebraic coefficients $\beta_0 + \beta_1 \log \alpha_1 + \cdots + \beta_n \log \alpha_n$ of logarithms of algebraic numbers [a3], a special case of which is Roy's strong six exponentials theorem: If $M$ is a $d \times \ell$-matrix, with $d\ell > d + \ell$, whose entries are linear combinations of logarithms of algebraic numbers, with rows linearly independent over the field of algebraic numbers and with columns linearly independent over the field of algebraic numbers, then the rank of $M$ is at least $2$.

An extension of Schneider's method also provides sharp measures for linear independence of logarithms of algebraic numbers [a1].

References

[a1]	M. Waldschmidt, "Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables" , Grundl. Math. Wissenschaft. , 326 , Springer (2000) MR1756786 Zbl 0944.11024
[a2]	S. Lang, "Introduction to transcendental numbers" , Addison-Wesley and Don Mills (1966) MR0214547 Zbl 0144.04101
[a3]	D. Roy, "Matrices whose coefficients are linear forms in logarithms" J. Number Theory , 41 : 1 (1992) pp. 22–47 MR1161143 Zbl 0763.11030
[a4]	K. Ramachandra, "Contributions to the theory of transcendental numbers I—II" Acta Arith. , 14 (1967/68) pp. 65–72; 73–88 MR0725957 MR0224566 Zbl 0176.33101
[a5]	Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen I" J. Reine Angew. Math. , 172 (1934) pp. 65–69 Zbl 0010.10501 Zbl 60.0163.03
[a6]	M. Waldschmidt, "Transcendance et exponentielles en plusieurs variables" Invent. Math. , 63 : 1 (1981) pp. 97–127 MR0608530 Zbl 0454.10020