The function compoly returns a pair ,  such that subs(x=q(x), p(x)) is equal to r, the input polynomial. If such a pair cannot be found, it returns FAIL.  and  are nonlinear polynomials and  has a low-degree in x greater or equal to 1.

When compoly is called without additional arguments, it is equivalent to being called with a second argument which is indets(r). When the second argument is a set with more than one variable, then a multivariate composition is attempted. In this case, the result has the same form, but the second polynomial is multivariate, that is, , .  For the multivariate case, the second polynomial, , may be of degree 1.

Note that the composition may not be unique. In particular, if ,  is a composition, then we can find another  such that replacing  by   will also result in a valid composition. This non-determinacy is eliminated by selecting c and b such that the q polynomial has integer content 1 and its independent term is 0.