Editing Transcendental number theory (section)

===Measure of transcendence of a complex number===
Next consider the values of polynomials at a complex number ''x'', when these polynomials have integer coefficients, degree at most ''n'', and [[Height of a polynomial|height]] at most ''H'', with ''n'', ''H'' being positive integers.

Let <math>m(x, n, H)</math> be the minimum non-zero absolute value such polynomials take at <math>x</math> and take:
:<math>\omega(x, n, H) = -\frac{\log m(x, n, H)}{n\log H}</math>
:<math>\omega(x, n) = \limsup_{H\to\infty}\, \omega(x,n,H).</math>

Suppose this is infinite for some minimum positive integer&nbsp;''n''.  A complex number ''x'' in this case is called a '''U&nbsp;number''' of degree&nbsp;''n''.

Now we can define 
:<math>\omega (x) = \limsup_{n\to\infty}\, \omega(x,n).</math> 
ω(''x'') is often called the '''measure of transcendence''' of&nbsp;''x''. If the ω(''x'', ''n'') are bounded, then ω(''x'') is finite, and ''x'' is called an '''S number'''.  If the ω(''x'', ''n'') are finite but unbounded, ''x'' is called a '''T number'''. ''x''&nbsp;is algebraic if and only if&nbsp;ω(''x'')&nbsp;=&nbsp;0.

Clearly the Liouville numbers are a subset of the U numbers.  [[William LeVeque]] in 1953 constructed U numbers of any desired degree.<ref name=LV172>{{harvnb|LeVeque|2002|p=II:172}}.</ref> The [[Liouville numbers]] and hence the U numbers are uncountable sets. They are sets of measure 0.<ref>{{harvnb|Burger|Tubbs|2004|p=170}}.</ref>

T numbers also comprise a set of measure 0.<ref>{{harvnb|Burger|Tubbs|2004|p=172}}.</ref> It took about 35 years to show their existence. [[Wolfgang M. Schmidt]] in 1968 showed that examples exist.  However, [[almost all]] complex numbers are S numbers.<ref name=Bug251/> Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:<ref>{{harvnb|LeVeque|2002|pp=II:174–186}}.</ref><ref>{{harvnb|Burger|Tubbs|2004|p=182}}.</ref> this shows that ''e'' is an S number and gives a proof of the transcendence of {{pi}}.  This number {{pi}} is known not to be a U number.{{sfn|Baker|1975|p=86}}  Many other transcendental numbers remain unclassified.

Two numbers ''x'', ''y'' are called '''algebraically dependent''' if there is a non-zero polynomial ''P'' in two indeterminates with integer coefficients such that ''P''(''x'',&nbsp;''y'')&nbsp;=&nbsp;0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.<ref name=LV172/><ref>{{harvnb|Burger|Tubbs|2004|p=163}}.</ref>  This allows construction of new transcendental numbers, such as the sum of a Liouville number with ''e'' or&nbsp;{{pi}}.

The symbol S probably stood for the name of Mahler's teacher [[Carl Ludwig Siegel]], and T and U are just the next two letters.