Real root finding for determinants of linear matrices

Let A0,A1,…,AnA0,A1,…,An be given square matrices of size m   with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety {x∈Rn:det⁡(A0+x1A1+⋯+xnAn)=0}{x∈Rn:det⁡(A0+x1A1+⋯+xnAn)=0}. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Under some genericity assumptions on the coefficients of the matrices, we provide an algorithm solving this problem whose runtime is essentially polynomial in the binomial coefficient (n+mn). We also report on experiments with a computer implementation of this algorithm. Its practical performance illustrates the complexity estimates. In particular, we emphasize that for subfamilies of this problem where m is fixed, the complexity is polynomial in n.