Computing bases of complete intersection rings in Noether position

Abstract

Let k be an effective infinite perfect field, k[ x 1,…, x n ] the polynomial ring in n variables and F∈ k[ x 1,…, x n ] M× M a square polynomial matrix verifying F 2= F. Suppose that the entries of F are polynomials given by a straight-line program of size L and their total degrees are bounded by an integer D. We show that there exists a well parallelizable algorithm which computes bases of the kernel and the image of F in time ( nL) O(1)( MD) O( n) . By means of this result we obtain a single exponential algorithm to compute a basis of a complete intersection ring in Noether position. More precisely, let f 1,…, f n− r ∈ k[ x 1,…, x n ] be a regular sequence of polynomials given by a slp of size ℓ, whose degrees are bounded by d. Let R≔ k[ x 1,…, x r ] and S≔ k[ x 1,…, x n ]/( f 1,…, f n− r ) such that S is integral over R; we show that there exists an algorithm running in time O( n)ℓ d O( n 2) which computes a basis of S over R. Also, as a consequence of our techniques, we show a single exponential well parallelizable algorithm which decides the freeness of a finite k[ x 1,…, x n ]-module given by a presentation matrix, and in the affirmative case it computes a basis.