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.