We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.

By adapting the effective version of Quillen-Suslin Theorem given in [8], we show that if the ideal defining a rational parametrization of degree d of an algebraic surface in 3-dimensional space is radical and has D points, then a μ-basis of this parametrization can be found of degree bounded by 5 max(1, D - 1)4 (2d + 1)4. This bound improves those obtained recently in [4] in our setup, and it is also sensitive to the number of base points.

We present a subresultant-based algorithm for deciding if the parametrization of a toric hypersurface is invertible or not, and for computing the inverse of the parametrization in the case where it exists. The algorithm takes into account the monomial structure of the input polynomials.

Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our MAPLE implementation, applied to certain examples, illustrates our construction and compares with alternative matrices.