A formal power series h(x) = sum a_n x^n is called algebraic if it satisfies a polynomial equation with coefficients in K[x]. Typical examples are rational series like (1+x)^(-1) or roots of polynomials like sqrt(1+x). In characteristic 0, it is an open problem to characterize the coefficient sequence (a_n) of an algebraic series, whereas in positive characteristic, the theory of automata provides a complete answer (which is, though, not entirely satisfactory).
Let h(x) be an algebraic series in one variable with rational coefficients a_n. Eisenstein observed that the denominators of a_n have only finitely many prime divisors, more precisely, that there exists a natural number d so that d^n a_n is an integer. This shows that exp(x) must be transcendental (= non algebraic), and, also, that the coefficients of an algebraic series tend to zero at most exponentially. But much more is true: if the set of coefficients is finite, h(x) is either a rational series or transcendental. In particular the lacunary series h(x) = sum x^(2^n) is transcendental (in characteristic different 2). Or: Every algebraic series is the diagonal (defined suitably) of a rational series in two variables. In the lecture, we will survey several of such spectacular and somewhat still mysterious results.
N.B. This talk is suitable for non-experts as well as students.