On the degrees of bases of free modules over a polynomial ring (Q1298001)

Let \(k\) be an infinite field, \(A\) the polynomial ring \(k[x_1,\dots, x_n]\) and \(F\in A^{N\times M}\) a matrix such that \(\operatorname {Im} F\subset A^N\) is \(A\)-free (in particular, the Quillen-Suslin theorem implies that \(\operatorname {Ker} F\) is also free). Let \(D\) be the maximum of the degrees of the entries of \(F\) and \(s\) the rank of \(F\). We show that there exists a basis \(\{v_1,\dots, v_M\}\) of \(A^M\) such that \(\{v_1,\dots, v_{M-s}\}\) is a basis of \(\operatorname {Ker}F\), \(\{F(v_{M-s+1}),\dots, F(v_M)\}\) is a basis of \(\operatorname {Im}F\) and the degrees of their coordinates are of order \(((M-s) sD)^{O(n^4)}\). This result allows to obtain a single exponential degree upper bound for a basis of the coordinate ring of a reduced complete intersection variety in Noether position.