On the degrees of bases of free modulesover a polynomial ring

Abstract

Let k be an infinite field, A the polynomial ring \(k[x_1,...,x_n]\) and \(F\in A^{N\times M}\) a matrix such that \({\rm Im}\,F\subset A^N\) is A-free (in particular, Quillen-Suslin Theorem implies that \({\rm Ker}\,F\) 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 \(\{ v_1,\ldots,v_{M} \}\) of \(A ^M\) such that \(\{ v_1,\ldots ,v_{M-s} \}\) is a basis of \({\rm Ker}\,F\), \(\{ F(v_{M-s+1}), \ldots , F(v_M) \}\) is a basis of \({\rm Im}\,F\) and the degrees of their coordinates are of order \(((M-s)sD)^{O(n^4)}\).