Real root finding for rank defects in linear Hankel matrices

Abstract: Let $H_0, ..., H_n$ be $m \times m$ matrices with entries in $\QQ$ and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix $H(\vecx)=H_0+\X_1H_1+...+\X_nH_n$ and the problem of computing sample points in each connected component of the real algebraic set defined by the rank constraint ${\sf rank}(H(\vecx))\leq r$, for a given integer $r \leq m-1$. Computing sample points in real algebraic sets defined by rank defects in linear matrices is a general problem that finds applications in many areas such as control theory, computational geometry, optimization, etc. Moreover, Hankel matrices appear in many areas of engineering sciences. Also, since Hankel matrices are symmetric, any algorithmic development for this problem can be seen as a first step towards a dedicated exact algorithm for solving semi-definite programming problems, i.e. linear matrix inequalities. Under some genericity assumptions on the input (such as smoothness of an incidence variety), we design a probabilistic algorithm for tackling this problem. It is an adaptation of the so-called critical point method that takes advantage of the special structure of the problem. Its complexity reflects this: it is essentially quadratic in specific degree bounds on an incidence variety. We report on practical experiments and analyze how the algorithm takes advantage of this special structure. A first implementation outperforms existing implementations for computing sample points in general real algebraic sets: it tackles examples that are out of reach of the state-of-the-art.