algebraic dependence of solution

lolo · New Member Offline Joined: 25 Oct 2004 Posts: 12

Hello,

Hope I am in the right section for this topic concerns with algebra (but related to number theory questions)

I should be very interested to know what is known about the following question or to have references about the following problem.

In a french "agregation" subject (highesht undergraduate exam) the purpose was to prove that solutions of the equation :

$f(x^2) = R(x) f(x)$ where R(x) is a rational function of complex variable , are rationals or hypertranscendentals (that means f and all it's derivatives are algebraically independent over C(z) ). I guess result may be extend to the case :
$f(x^d) = R(x) f(x)$ (d integer>1)

Please if you know: a) what happen if the complex field is replaced by a field of characteristic p .
b) what happen (in all characteristics) for equation

$f(x^{d^2}) = R(x) f(x^d) + S(x)f(x)$ ( R and S rationnal)

c) what about inhomogenous equations $f(x^2) = R(x) f(x) +c(x)$ ?

Thanks very much for any reference (including the first type of equation because I do not know which has proved this result the first time)

**Posted:** Sun Oct 31, 2004 1:10 am