A real polynomial is hyperbolic whenever all its roots are real or, equivalently, if it is the characteristic polynomial of a real symmetric matrix. Testing efficiently this property, that can be extended to the multivariate case, is an open problem in computer science and mathematics. Whereas hyperbolic polynomials constitute nowadays a central topic in the applications, the computational/complexity aspects have a limited role in this theory. The general goal of HYPERSPACE is to develop an effective approach to hyperbolic polynomials. On the one hand, we propose to improve existing algorithms for the computation of determinantal representations (a hyperbolicity certificate that exists in some cases) or independent of classical representations. On the other hand, the project will be consecrated to the implementation of new algorithms in a free software dedicated to hyperbolic polynomials.