Sumario: | Let k be an infinite field, A the polynomial ring k[x1 , . . . , xn] and F ∈ AN×M a matrix such that Im F ⊂ AN A-free (in particular, Quillen-Suslin Theorem implies that 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 {v1 , . . . , vM} of AM such that {v1 , . . . , vM-s} is a basis of Ker F, {F(vM-s+1), . . . , F(vM)} is a basis of Im F and the degrees of their coordinates are of order ((M - s)sD)O(n4). 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.
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