On the closed image of a rational map and the implicitization problem

In this paper, we investigate some topics around the closed image $S$ of a rational map $\lambda$ given by some homogeneous elements $f_1,...,f_n$ of the same degree in a graded algebra $A$. We first compute the degree of this closed image in case $\lambda$ is generically finite and $f_1,...,f_n$ define isolated base points in \Proj(A). We then relate the definition ideal of $S$ to the symmetric and the Rees algebras of the ideal $I=(f_1,...,f_n) \subset A$, and prove some new acyclicity criteria for the associated approximation complexes. Finally, we use these results to obtain the implicit equation of $S$ in case $S$ is a hypersurface, \Proj(A)=\PP^{n-2}_k with $k$ a field, and base points are either absent or local complete intersection isolated points.Comment: 43 pages, revised version. To appear in Journal of Algebr

Bridging the Gap Between Ox and Gauss using OxGauss

The purpose of this paper is to review and discuss the key improvements brought to OxGauss. Without having to install Gauss on his or her machine, the OxGauss user can run under Ox a wide range of Gauss programs and codes. Even with the console Ox version (free for academics), Gauss codes can either be called from Ox programs or run and executed on their own. While the new OxGauss version is very powerful in most circumstances, it is of little use once the purpose is to execute programs that attempt to solve optimization problems using Cml, Maxlik or Optmum. In this paper we propose a set of additional procedures that contribute to bridge the gap between Ox and three well-known Gauss application modules: Cml, Maxlik or Optmum.The effectiveness of our procedures is illustrated by revisiting a large number of freely available Gauss codes in which numerical optimization relies on the above Gauss application modules. The Gauss codes include many programs dealing with non-linear models such as the Markov regime-switching models STAR models and various GARCH-type models. These illustrations highlight a further potentially interesting implication of OxGauss: it enables non-Gauss users to replicate existing empirical results using freely available Gauss codes.econometrics;