Computing bases of complete intersection rings in Noether position

Let k be an effective infinite perfect field, k[x1,…,xn] the polynomial ring in n variables and Fset membership, variantk[x1,…,xn]M×M a square polynomial matrix verifying F2=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 f1,…,fn−rset membership, variantk[x1,…,xn] be a regular sequence of polynomials given by a slp of size ℓ, whose degrees are bounded by d. Let Rcolon, equalsk[x1,…,xr] and Scolon, equalsk[x1,…,xn]/(f1,…,fn−r) such that S is integral over R; we show that there exists an algorithm running in time O(n)ℓdO(n2) 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[x1,…,xn]-module given by a presentation matrix, and in the affirmative case it computes a basis.