Abstract We generalize Kedlaya and Umans' modular composition algorithm to the
multivariate case. As a main application, we give fast algorithms for many operations
involving triangular sets (over a finite field), such as modular multiplication, inversion, or
change of order. For the first time, we are able to exhibit running times for these operations
that are almost linear, without any overhead exponential in the number of variables. As a
further application, we show that, from the complexity viewpoint, Charlap, Coley, and …