Real factorization of multivariate polynomials with integer coefficients

In the recent works ([4, 15]), a new efficient probabilistic semi-numerical absolute (i.e. complex) factorization algorithm for multivariate polynomials with integer coefficients is given. It was based on a simple property of the monomials appearing after a generic linear change of coordinates for bivariate polynomials and a deep result of complex algebraic geometry.
Here we consider the a priori simpler problem of factorization over the field of real numbers. We briefly review our algorithm for complex factorization and adapt it to solve the problem on the reals. This allows to spare a significant part of the computations and improve the range of tractability. The method provides factors with approximative coefficients and eventually exact factors in a suitable real algebraic extension of the field $\mathbb Q$.