Suslin's algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring A if (v"1(X),...,v"n(X))@?(A[X])^n is unimodular where v"1 is monic and n>=3, then there exist @c"1,...,@c"@?@?E"n"-"1(A[X]) such that the ideal generated by Res(v"1,e"1.@c"1^t(v"2,...,v"n)),...,Res(v"1,e"1.@c"@?^t(v"2,...,v"n)) equals A. This lemma played a central role in the resolution of Serre's Conjecture. In the case where A contains a set E of cardinality greater than degv"1+1 such that y-y^' is invertible for each y y^' in E, we prove that the @c"i can simply correspond to the elementary operations L"1->L"1+y"i@?"j"="2^n^-^1u"j"+"1L"j, 1@?i@?@?=degv"1+1, where u"1v"1+...+u"nv"n=1. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in K[X"1,...,X"k] to ^t(1,0,...,0) using elementary operations in the case where 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.