Algorithms for computing greatest common divisors of parametric multivariate polynomials

Two new efficient algorithms for computing greatest common divisors (gcds) of parametric multivariate polynomials over k[U][X] are presented. The key idea the first algorithm is that gcd two non-parametric can be obtained by dividing their product generator intersection principal ideals generated polynomials. second based on another simple insight extracted using ideal quotient a polynomial with respect to polynomial. Since and in these cases also ideals, generators minimal Gröbner bases quotient, respectively. To avoid introducing variables which adversely affect efficiency, computations performed modules. Both constructions generalize case as shown paper. Comprehensive system used Kapur-Sun-Wang's algorithm. It proved whether comprehensive or each branch specializations corresponds single generator. Using this generator, division. For more than polynomials, we use above compute gcds recursively, get an extended generalizing Algorithms do not suffer from having apply expensive steps such ensuring primitive w.r.t. main variable both proposed Nagasaka (ISSAC, 2017). resulting only conceptually understand but practice. Nagasaka's have been implemented Singular, performance compared number examples.