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.