The purpose of this article is to present some recent applications of computer algebra to answer structural and numerical questions in applied sciences. A first example concerns identifiability which is a pre-condition for safely running parameter estimation algorithms and obtaining reliable results. Identifiability addresses the question whether it is possible to uniquely estimate the model parameters for a given choice of measurement data and experimental input. As discussed in this paper, symbolic computation offers an efficient way to do this identifiability study and to extract more information on the parameter properties. A second example addressed hereafter is the diagnosability in nonlinear dynamical systems. The diagnosability is a prior study before considering diagnosis. The diagnosis of a system is defined as the detection and the isolation of faults (or localization and identification) acting on the system. The diagnosability study determines whether faults can be discriminated by the mathematical model from observations. These last years, the diagnosability and diagnosis have been enhanced by exploitting new analytical redundancy relations obtained from differential algebra algorithms and by the exploitation of their properties through computer algebra techniques.

In this article, we present new results about the computation of a general shape of a triangular basis generating the splitting ideal of an irreducible polynomial given with the permutation representation of its Galois group G. We provide some theoretical results and a new general algorithm based on the study of the non redundant bases of permutation groups. These new results deeply increase the efficiency of the computation of the splitting field of a polynomial.