Algorithmic aspects of Suslin's proof of Serre's conjecture

On the degrees of bases of free modulesover a polynomial ring

“…. , x n−1 )[x n ] -unimodular matrix in order to apply Suslin's reduction procedure following [17] and [5] (see the next two lemmas). Unfortunately this approach requires the introduction of certain polynomials in B playing the rôle of the ξ 's.…”

“…For example, if the matrix is unimodular (i.e. the rows can be extended to a basis of the whole space), a single exponential upper bound for the degree of a basis of its kernel is given in [5,Corollary 3.2].…”

“…2 to 5) are strongly inspired on the works of D.Quillen, A.Suslin, L.Vaserstein and M.Hochster related to the resolution of the so called "Serre's Conjecture" (on this subject let us mention the remarkable books of T.Y.Lam [17] and E.Kunz [15]). We combine this approach with the effective version of Hilbert Nullstellensatz (see [13], [9] and the references given in [3] and [22]) and its consequences in the quantitative study of polynomial unimodular matrices following [5]. Another approach on effective Quillen-Suslin Theorem by means of Gröbner Basis may be found in [18].…”

An algorithm for unimodular completion over Noetherian rings

“…, X k ]. The only precise complexity bounds for such problems we found in the literature are in case A is a field and are given in [4,6,19]. The best bounds are given in [4] while the main feature of the algorithm given in [19] is its simplicity and the fact that it considerably reduces the number of the utilized elementary matrices (see the remainder hereafter about elementary matrices).…”

Generic Invertibility of Multidimensional FIR Filter Banks and MIMO Systems

“…Caniglia et al [28] propose an upper bound on the degree of N ×N invertible matrix K(z) such that K(z)H(z) = I P 0 and the degree bound of deg(K(z)) is optimal in order. Proposition 3: [28] Assume that H(z) is an N × P invertible matrix in M variables. Let deg(H(z)) be the maximum of the degrees of the entries of H(z) and let…”