Simultaneous Elimination by using Several Tools from Real Algebraic Geometry

The aim of this paper is to introduce two new elimination procedures for algebraic systems of equations. The first one eliminates one variable from a finite set of polynomials with complex or real coefficients and it is based on a parametric version of Barnett’s Method for computing the greatest common divisor of a finite family of univariate polynomials. The second one, based on Hermite’s Method, deals with the global elimination of a block of variables from a finite set of multivariate polynomials with a particular structure (containing a Pham system). A common feature of both procedures is that the final step relies on a specific property of a real-valued inner product on vector spaces over the coefficient field: Gram’s Criterion.