View source for Siegel method

A method for investigating the arithmetical properties of the values assumed at algebraic points by [[E-function|$E$-function]]s that satisfy linear differential equations with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850002.png" />; first proposed by C.L. Siegel [[#References|[1]]].

An entire function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850003.png" /></td> </tr></table>

is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850005.png" />-function if all its coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850006.png" /> belong to an algebraic field of finite degree (cf. [[Algebraic number|Algebraic number]]; [[Field|Field]]), if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850007.png" /> the maximum modulus of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850008.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s0850009.png" /> and if there exists a sequence of rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500011.png" /> is an algebraic integer for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500012.png" />. Examples are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500014.png" /> and the Bessel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500015.png" /> (cf. [[Bessel functions|Bessel functions]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500020.png" /> are rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500022.png" />, then the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500023.png" /></td> </tr></table>

is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500024.png" />-function; it satisfies a linear differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500025.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500026.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500027.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500029.png" /> is the Bessel function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500030.png" /> is a rational number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500031.png" /> then for any algebraic number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500032.png" /> the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500034.png" /> are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500035.png" /> (cf. [[Algebraic independence|Algebraic independence]]).

In 1949 Siegel presented his method in a general setting, but the conditions that had to be imposed on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500036.png" />-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500037.png" /> 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 [[#References|[2]]]–[[#References|[3]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500038.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500039.png" />-functions which constitute a solution to the system of differential equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500041.png" /> be an algebraic number distinct from zero and from the singular points of the system (1); then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500042.png" /> numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500043.png" /> are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500044.png" /> if and only if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500045.png" /> are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500046.png" />. This theorem implies, in particular, that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500047.png" /> are algebraically independent, then all the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500048.png" /> are transcendental (cf. [[Transcendental number|Transcendental number]]); the same holds for all non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500049.png" />-points of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500050.png" /> distinct from the poles of the system (1), provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500051.png" /> is algebraic. The theorem has produced a great number of results concerning specific <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500052.png" />-functions, and algebraic independence proofs for values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500053.png" />-functions satisfying linear homogeneous and inhomogeneous differential equations of order higher than two. For example, the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500054.png" /></td> </tr></table>

satisfies a linear differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500055.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500056.png" />; it can be proved that for any algebraic number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500057.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500058.png" /> numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500060.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500061.png" />, are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500062.png" />.

Under the same conditions, the maximum number of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500063.png" /> which are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500064.png" /> is equal to the maximum number of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500065.png" /> which are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500067.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500068.png" />-functions that are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500069.png" /> and that satisfy the system (1), then for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500070.png" />, with the possible exception of finitely many, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500071.png" /> are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500072.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500073.png" />-functions at algebraic points.

Siegel's methods enables one to estimate the measure of algebraic independence of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500074.png" />, thus giving the results a quantitative form. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500075.png" /> are algebraically independent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500077.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500079.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500080.png" /> and the degree of the algebraic number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500081.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.L. Siegel,   "Ueber einige Anwendungen Diophantischer Approximationen"  ''Abh. Deutsch. Akad. Wiss. Phys.- Math. Kl.'' :  1  (1929)  pp. 1–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.B. Shidlovskii,   "On transcendency and algebraic independence of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500082.png" />-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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Lang,   "A transcendence measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500083.png" />-functions"  ''Mathem.'' , '''9'''  (1962)  pp. 157–161</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
In the article above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500084.png" /> is the measure of algebraic independence, cf. [[Algebraic independence, measure of|Algebraic independence, measure of]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.B. Shidlovskii,   "Transcendental numbers" , de Gruyter  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. André,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085000/s08500085.png" />-functions" , Vieweg  (1989)</TD></TR></table>