Determinantal systems are systems of polynomial equations which encode a rank deficiency of a given matrix with polynomial entries over the solution set to other polynomial equations. Such systems arise in a number of areas of computational mathematics such as polynomial optimization, real algebraic and enumerative geometry and engineering sciences such as robotics and biology.

In this talk, I will describe some structural results on the ideals generated by such systems. These lead to new algorithms to compute Grobner bases for such ideals that are specifically tuned to solve determinantal systems more efficiently. Grobner bases are special finite sets of generators which can be used to solve the ideal membership problem and to extract the roots of the ideal at hand. As is the case for state of the art Grobner basis algorithms, these new algorithms are based on linearization techniques, which originate with the work of Macaulay on multivariate resultants. We show how to reduce the size of the matrices which arise in this context as well as use cases where the matrices built by our algorithms are full rank, which lead to lower bounds on the amount of data that must be handled by such algorithms. This is joint work with Vincent Neiger and Mohab Safey El Din.