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In 1934 Hilbert's seventh problem (cf. also [[Hilbert problems]]) was solved independently by A.O. Gel'fond [[#References|[a3]]] and Th. Schneider [[#References|[a8]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300201.png" /> is a non-zero [[algebraic number]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300202.png" /> a non-zero logarithm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300204.png" /> an irrational algebraic number, then the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300205.png" /> is transcendental (cf. [[Transcendental number|Transcendental number]]).
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The transcendence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300206.png" /> (corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300209.png" />) had already been proved by Gel'fond in 1929 [[#References|[a2]]] using interpolation formulas for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002010.png" />, like in Pólya's work [[#References|[a7]]] on integral-valued entire functions.
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In 1934 Hilbert's seventh problem (cf. also [[Hilbert problems]]) was solved independently by A.O. Gel'fond [[#References|[a3]]] and Th. Schneider [[#References|[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|Transcendental number]]).
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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 [[#References|[a2]]] using interpolation formulas for the function $e ^ { \pi z }$, like in Pólya's work [[#References|[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|Dirichlet principle]]).
 
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|Dirichlet principle]]).
  
While Schneider's proof (cf. [[Schneider method|Schneider method]]) is based on the addition theorem for the exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002011.png" />, the main ingredient in Gel'fond's proof is the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002012.png" />. Gel'fond considers the two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002014.png" />; his auxiliary function has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002016.png" /> is a polynomial with algebraic coefficients. He investigates the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002017.png" /> as well as its derivatives at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002019.png" />. An extrapolation is an essential feature of his proof.
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While Schneider's proof (cf. [[Schneider method|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 [[#References|[a4]]]; see also [[Gel'fond–Baker method]]; [[Diophantine approximations|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 [[#References|[a9]]] and of Abelian integrals [[#References|[a10]]], including the transcendence of the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002020.png" /> of the beta-function at rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002021.png" />. Next, Schneider [[#References|[a11]]], [[#References|[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 [[#References|[a5]]], who extended Schneider's results to commutative algebraic groups. The following far-reaching statement is called the Schneider–Lang criterion: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002022.png" /> be a [[Number field|number field]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002023.png" /> be meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002024.png" /> of finite order of growth (cf. also [[Meromorphic function|Meromorphic function]]). Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002026.png" /> are algebraically independent (cf. also [[Algebraic independence|Algebraic independence]]). Assume also that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002027.png" />, the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002029.png" /> belongs to the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002030.png" />. Then the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002031.png" /> that are not poles of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002032.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002033.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002034.png" />, is finite.
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This method has been developed by Gel'fond himself for proving quantitative Diophantine approximation estimates (see [[#References|[a4]]]; see also [[Gel'fond–Baker method]]; [[Diophantine approximations|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 [[#References|[a9]]] and of Abelian integrals [[#References|[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 [[#References|[a11]]], [[#References|[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 [[#References|[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|number field]] and let $f _ { 1 } , \ldots , f _ { d }$ be meromorphic functions in $\mathbf{C}$ of finite order of growth (cf. also [[Meromorphic function|Meromorphic function]]). Assume $f _ { 1 }$, $f _ { 2 }$ are algebraically independent (cf. also [[Algebraic independence|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 [[#References|[a5]]], has been obtained by E. Bombieri [[#References|[a1]]].
 
Schneider and Lang extended their criterion to several variables by considering Cartesian products; a deeper result, involving algebraic hypersurfaces and suggested by M. Nagata [[#References|[a5]]], has been obtained by E. Bombieri [[#References|[a1]]].
  
A clever modification of the Gel'fond–Schneider method has been applied to modular functions in [[#References|[a13]]], solving Mahler's conjecture: For any algebraic number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002035.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002036.png" /> the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002037.png" /> of the [[modular function]] is transcendental.
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A clever modification of the Gel'fond–Schneider method has been applied to modular functions in [[#References|[a13]]], solving Mahler's conjecture: For any algebraic number $\alpha$ with $0 &lt; | \alpha | &lt; 1$ the value $J ( \alpha )$ of the [[modular function]] is transcendental.
  
Gel'fond proved in 1949 the algebraic independence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002039.png" />. More generally, he proved that for algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002041.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002043.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002044.png" />, the transcendence degree over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002045.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002046.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002047.png" /> (cf. also [[Transcendental extension|Transcendental extension]]). After the work of G.V. Chudnovskii, P. Philippon and G. Diaz, it is known that this transcendence degree is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002048.png" />. This method not only provides a new proof of the [[Lindemann–Weierstrass theorem]] on the algebraic independence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002049.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002050.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002051.png" />-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.
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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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002053.png" /> (showing therefore that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002054.png" /> is transcendental), and later Yu.V. Nesterenko adapted the method of [[#References|[a13]]] and obtained remarkable results of algebraic independence on values of modular functions, including the algebraic independence of the three numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g13002057.png" /> [[#References|[a6]]].
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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 [[#References|[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 $e ^ { \pi }$ [[#References|[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]]).
 
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Bombieri, "Algebraic values of meromorphic maps" ''Invent. Math.'' , '''10''' (1970) pp. 267–287 (Addendum, 11 (1970), 163-166)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.O. Gel'fond, "Sur les propriétés arithmétiques des fonctions entières" ''Tôhoku Math. J.'' , '''30''' (1929) pp. 280–285</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover (1960) (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Lang, "Introduction to transcendental numbers" , Addison-Wesley and Don Mills (1966) (reprinted in: Collected Papers, Vol. I, Springer, 2000, pp. 396-506) {{MR|0214547}} {{ZBL|0144.04101}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> "Introduction to algebraic independence theory. Instructional Conference (CIRM Luminy, 1997)" Y.V. Nesterenko (ed.) P. Philippon (ed.) , ''Lecture Notes in Mathematics'' , '''1752''' , Springer (2001)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen I" ''J. Reine Angew. Math.'' , '''172''' (1934) pp. 65–69 {{MR|}} {{ZBL|0010.10501}} {{ZBL|60.0163.03}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen II" ''J. Reine Angew. Math.'' , '''172''' (1934) pp. 70–74</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> Th. Schneider, "Zur Theorie der Abelschen Funktionen und Integrale" ''J. Reine Angew. Math.'' , '''183''' (1941) pp. 110–128</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> Th. Schneider, "Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise" ''Math. Ann.'' , '''121''' (1949) pp. 131–140</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> Th. Schneider, "Einführung in die transzendenten Zahlen" , Springer (1957)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> 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</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> E. Bombieri, "Algebraic values of meromorphic maps" ''Invent. Math.'' , '''10''' (1970) pp. 267–287 (Addendum, 11 (1970), 163-166)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> A.O. Gel'fond, "Sur les propriétés arithmétiques des fonctions entières" ''Tôhoku Math. J.'' , '''30''' (1929) pp. 280–285</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover (1960) (In Russian)</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> S. Lang, "Introduction to transcendental numbers" , Addison-Wesley and Don Mills (1966) (reprinted in: Collected Papers, Vol. I, Springer, 2000, pp. 396-506) {{MR|0214547}} {{ZBL|0144.04101}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> "Introduction to algebraic independence theory. Instructional Conference (CIRM Luminy, 1997)" Y.V. Nesterenko (ed.) P. Philippon (ed.) , ''Lecture Notes in Mathematics'' , '''1752''' , Springer (2001)</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> 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)</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen I" ''J. Reine Angew. Math.'' , '''172''' (1934) pp. 65–69 {{MR|}} {{ZBL|0010.10501}} {{ZBL|60.0163.03}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> Th. Schneider, "Transzendenzuntersuchungen periodischer Funktionen II" ''J. Reine Angew. Math.'' , '''172''' (1934) pp. 70–74</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> Th. Schneider, "Zur Theorie der Abelschen Funktionen und Integrale" ''J. Reine Angew. Math.'' , '''183''' (1941) pp. 110–128</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> Th. Schneider, "Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise" ''Math. Ann.'' , '''121''' (1949) pp. 131–140</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> Th. Schneider, "Einführung in die transzendenten Zahlen" , Springer (1957)</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> 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</td></tr></table>

Revision as of 16:58, 1 July 2020

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
How to Cite This Entry:
Gel'fond-Schneider method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gel%27fond-Schneider_method&oldid=50279
This article was adapted from an original article by Michel Waldschmidt (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article