Dimension-free matricial Nullstellens\"atze for noncommutative polynomials

Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry, and has inspired a plethora of noncommutative analogs. In last two decades, there has been an increased interest in understanding vanishing sets of polynomials in several matrix variables without restricting the matrix size, prompted by developments in noncommutative function theory, control systems, operator algebras, and quantum information theory. The emerging results vary according to the interpretation of what vanishing means. For example, given a collection of noncommutative polynomials, one can consider all matrix tuples at which the values of these polynomials are all zero, singular, have common kernel, or have zero trace. This survey reviews Nullstellens\"atze for the above types of vanishing sets, and identifies their structural counterparts in the free algebra.