The command PreComprehensiveTriangularize(sys, d, R)  returns a pre-comprehensive triangular decomposition of sys, with respect to the last d variables of R.

A pre-comprehensive triangular decomposition is a refined triangular decomposition (in the Lazard sense) with additional properties, aiming at studying parametric polynomial systems.

Let  be the last d variables of R, which we regard as parameters. A finite set  of regular chains of R forms a pre-comprehensive triangular decomposition of F with respect to U, if for every parameter value , there exists a subset  of  such that

(1) the regular chains of  specialize well at , and

(2) after specialization at , these chains form a triangular decomposition (in the Lazard sense) of the polynomial system  specialized at . See the command DefiningSet for the term specialize well.