The command BivariateModularTriangularize(F, R) returns a triangular decomposition of F in R. See the command Triangularize and the page RegularChains for the concept of a triangular decompostion.

F consists of two bivariate polynomials f1 and f2 of R. No other assumptions are required.

R must have only two variables and no parameters.

Moreover R must have a prime characteristic  such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f1 and f2 are, the larger must be  such that  divides .  If the degree of  f1 or f2 is too large, then an error is raised.

The algorithm is deterministic (i.e. non-probabilistic) and uses modular techniques together with asymptotically fast polynomial arithmetic.

When both Triangularize and BivariateModularTriangularize apply, the latter command is very likely to outperform the former one.