Suslin's algorithms for reduction of unimodular rows

with Ihsen Yengui
Journal of Symbolic Computation 39 (2005), 707-717.
Suslin's algorithms for reduction of unimodular rows. fichier pdf

Abstract :

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.