  A new approach is established to computing the image of a rational map,
whereby the use of approximation complexes is complemented with a detailed
analysis of the torsion of the symmetric algebra in certain degrees. In the
case the map is everywhere defined this analysis provides free resolutions of
graded parts of the Rees algebra of the base ideal in degrees where it does not
coincide with the corresponding symmetric algebra. A surprising fact is that
the torsion in those degrees only contributes to the first free module in the
resolution of the symmetric algebra modulo torsion. An additional point is that
this contribution -- which of course corresponds to non linear equations of the
Rees algebra -- can be described in these degrees in terms of non Koszul
syzygies via certain upgrading maps in the vein of the ones introduced earlier
by J. Herzog, the third named author and W. Vasconcelos. As a measure of the
reach of this torsion analysis we could say that, in the case of a general
everywhere defined map, half of the degrees where the torsion does not vanish
are understood.
