Section Resultant.
Variables (R : ringType) (p q : {poly R}).
Let dS := ((size q).-1 + (size p).-1)%N.
Local Notation band r := (lin1_mx (poly_rV \o r \o* rVpoly)).
Definition Sylvester_mx : 'M[R]_dS := col_mx (band p) (band q).
Definition resultant := \det Sylvester_mx.
End Resultant.
Prenex Implicits Sylvester_mx resultant.

Lemma resultant_in_ideal (R : comRingType) (p q : {poly R}) :
size p > 1 -> size q > 1 ->
{uv : {poly R} * {poly R} | size uv.1 < size q /\ size uv.2 < size p
& (resultant p q)%:P = uv.1 * p + uv.2 * q}.