  Discrete Differential Equations (DDEs) are functional equations that relate
polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a
"catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and
of some of its partial derivatives in $u$. DDEs occur frequently in
combinatorics, especially in map enumeration. If a DDE is of fixed-point type
then its solution $F(t,u)$ is unique, and a general result by Popescu (1986)
implies that $F(t,u)$ is an algebraic power series. Constructive proofs of
algebraicity for solutions of fixed-point type DDEs were proposed by
Bousquet-M\'elou and Jehanne (2006). Bostan et. al (2022) initiated a
systematic algorithmic study of such DDEs of order 1.
  We generalize this study to DDEs of arbitrary order. First, we propose
nontrivial extensions of algorithms based on polynomial elimination and on the
guess-and-prove paradigm. Second, we design two brand-new algorithms that
exploit the special structure of the underlying polynomial systems. Last, but
not least, we report on implementations that are able to solve highly
challenging DDEs with a combinatorial origin.

Symbolic Computation;Algebraic Geometry;Combinatorics