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 $((M-s)sD)^{O(n^4)}$.

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.