Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x]

Deterministic algorithms are given for some computational problems that take as input a nonsingular polynomial matrix A over K[x], K an abstract field, including solving a linear system involving A and computing a row reduced form of A. The fastest known algorithms for linear system solving based on the technique of high-order lifting by Storjohann (2003), and for row reduction based on fast minimal approximant basis computation algorithm by Giorgi, Jeannerod and Villard (2003), use randomization to find either a linear or small degree polynomial that is relatively prime to detA. We derandomize these algorithms by first computing a factorization of A = UH, with x not dividing detU and x − 1 not dividing detH. A partial linearization technique, that is applicable also to other problems, is developed to transform a system involving H, which may have some columns of large degrees, to an equivalent system that has degrees reduced to that of the average column degree.