A well-known lemma of Suslin says that for a commutative ring A if (v1(X), . . .,vn(X)) is a unimodular vector where v1 is monic and n >2, then there exist gamma1, . . .,gammaell in En-1(A[X]) such that
(Res(v1, e1.gamma1 (v2, . . .,vn)T), . . ., Res (v1, e1.gammaell (v2, . . ., vn)T) = A .
This lemma played a central role in the resolution of Serre's conjecture.
In case A contains a set E of cardinal greater than deg v1+1 such that y-y' is invertible for each pair y , y' of distinct elements in E , we prove that the gammai can simply correspond to the elementary operations
L1 <-- L1 + yi sumj=2n-1 uj+1Lj , (0 < i < ell+1= deg v1+2),
where u1v1+ . . .+unvn =1.
These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in K[X1, . . .,Xk] to (1,0, . . .,0)T using elementary operations in case K is an infinite field.
Another feature of this paper is that it shows that the concrete local-global principles can produce competitive complexity bounds.