  In this contribution, we consider a zero-dimensional polynomial system in $n$
variables defined over a field $\mathbb{K}$. In the context of computing a
Rational Univariate Representation (RUR) of its solutions, we address the
problem of certifying a separating linear form and, once certified, calculating
the RUR that comes from it, without any condition on the ideal else than being
zero-dimensional. Our key result is that the RUR can be read (closed formula)
from lexicographic Groebner bases of bivariate elimination ideals, even in the
case where the original ideal that is not in shape position, so that one can
use the same core as the well known FGLM method to propose a simple algorithm.
Our first experiments, either with a very short code (300 lines) written in
Maple or with a Julia code using straightforward implementations performing
only classical Gaussian reductions in addition to Groebner bases for the degree
reverse lexicographic ordering, show that this new method is already
competitive with sophisticated state of the art implementations which do not
certify the parameterizations.
