We give a new structure theorem for subresultants precising their
gap structure and derive from
it a new algorithm for computing them.
If d is a bound on the degrees and t a bound on the bitsize of the minors extracted from Sylvester matrix, our algorithm has O(d^2) arithmetic operations and size of intermediate computations 2t.
The key idea is to precise the relations between the successive Sylvester submatrices of A and B in one hand and of A and B on the other hand, using the notion of G-remainder we introduce.