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Control theory has recently been involved in the field of nuclear magneticresonance imagery. The goal is to control the magnetic field optimally in orderto improve the contrast between two biological matters on the pictures.Geometric optimal control leads us here to analyze mero-morphic vector fieldsdepending upon physical parameters , and having their singularities defined bya deter-minantal variety. The involved matrix has polynomial entries withrespect to both the state variables and the parameters. Taking into account thephysical constraints of the problem, one needs to classify, with respect to theparameters, the number of real singularities lying in some prescribedsemi-algebraic set. We develop a dedicated algorithm for real rootclassification of the singularities of the rank defects of a polynomial matrix,cut with a given semi-algebraic set. The algorithm works under some genericityassumptions which are easy to check. These assumptions are not so restrictiveand are satisfied in the aforementioned application. As more general strategiesfor real root classification do, our algorithm needs to compute the criticalloci of some maps, intersections with the boundary of the semi-algebraicdomain, etc. In order to compute these objects, the determinantal structure isexploited through a stratifi-cation by the rank of the polynomial matrix. Thisspeeds up the computations by a factor 100. Furthermore, our implementation isable to solve the application in medical imagery, which was out of reach ofmore general algorithms for real root classification. For instance,computational results show that the contrast problem where one of the mattersis water is partitioned into three distinct classes.

Determinantal sets, singularities and application to optimal control in medical imagery