Suslin’s algorithms for reduction of unimodular rows

AbstractA well-known lemma of Suslin says that for a commutative ring A if (v1(X),…,vn(X))∈(A[X])n is unimodular where v1 is monic and n≥3, then there exist γ1,…,γℓ∈En−1(A[X]) such that the ideal generated by Res(v1,e1.γ1t(v2,…,vn)),…,Res(v1,e1.γℓt(v2,…,vn)) 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 degv1+1 such that y−y′ is invertible for each y≠y′ in E, we prove that the γi can simply correspond to the elementary operations L1→L1+yi∑j=2n−1uj+1Lj, 1≤i≤ℓ=degv1+1, 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 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