We study the coefficients of algebraic functions
∑n≥0fnzn. First, we recall the too-little-known fact that these coefficients
fn always admit a closed form. Then we study their asymptotics, known to be
of the type fn ~ CAnnα. When the function is a power series
associated to a context-free grammar, we solve a folklore conjecture: the
critical exponents α cannot be 1/3 or −5/2; they in fact
belong to a proper subset of the dyadic numbers. We initiate the study of the
set of possible values for A. We extend what Philippe Flajolet
called the Drmota–Lalley–Woods theorem (which states that
α=−3/2 when the dependency graph associated to the
algebraic system defining the function is strongly connected). We fully
characterize the possible singular behaviours in the non-strongly connected
case. As a corollary, the generating functions of certain lattice paths and
planar maps are not determined by a context-free grammar (i.e.,
their generating functions are not ℕ-algebraic). We give examples of
Gaussian limit laws (beyond the case of the
Drmota–Lalley–Woods theorem), and examples of non-Gaussian
limit laws. We then extend our work to systems involving non-polynomial entire
functions (non-strongly connected systems, fixed points of entire functions with
positive coefficients). We give several closure properties for
ℕ-algebraic functions. We end by discussing a few extensions of our
results (infinite systems of equations, algorithmic aspects).