A new algorithm for the Quillen-Suslin theorem (Q1975382)

Proceeding from their more general constructions [\textit{R. C. Laubenbacher} and \textit{C. J. Woodburn}, J. Pure Appl. Algebra 117-118, 395-429 (1997; Zbl 0887.13006)], the authors propose a new algorithm for a part of the inductive algorithm of \textit{A. Logar} and \textit{B. Sturmfels} [J. Algebra 145, No. 1, 231-239 (1992; Zbl 0747.13020)]. Namely, let \(K\) be a field, \(P\) be a finitely generated module over the ring of polynomials \(K[x_1,x_2, \dots, x_r,y]\), given as the kernel or the cokernel of a matrix with entries of this ring, \(M\) be a maximal ideal of \(K[x_1,x_2, \dots,x_r]\), \(P_M\) be the corresponding localized module over the ring \(K[x_1,x_2, \dots, x_r]_M[y]\). The algorithm decides, whether \(P_M\) is projective and computes a free basis for \(P_M\) in this case.