| Résume | [a report on on-going joint work with Alice Medvedev and Khoa Nguyen]
In the 1930s, Mahler developed a method for proving the transcendence of special values of certain analytic functions by using the functional equations satisfied by these functions. In recent years, the difference Galois theory has been used to study the algebraic relations on Mahler functions satisfying linear difference equations. I will talk about Mahler functions satisfying nonlinear equations. More specifically, given a natural number k > 1 and a Laurent series f(x) in C((x)) (where C is an algebraically closed field of characteristic zero), we say that f is a k-Mahler function if there is a rational function P(x,y) which is a polynomial of degree at least 2 in y for which f satisfies the functional equation f(x^k) = P(x,f(x)). Zannier has shown that if f is a k-Mahler function which is algebraic over C(x), then f in C(x). We study the following question: If f and g are non-rational k-Mahler and l-Mahler functions, respectively, with k and l multiplicatively independent, must f and g be algebraically independent over C(x)?
Using our theory of σ-degree one difference varieties defined by polynomials, we reduce the problem to an apparently simpler problem of skew-conjugation between Galois-conjugate difference equations.
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