Strength and polynomial functors

One of the most important invariants one can associate to a matrix is its rank, which expresses how many pairs of vectors you need to write down a formula for the matrix. Consider infinite-by-infinite matrices. For such a matrix, we define its rank to be the supremum of all its finite-by-finite submatrices. This can be finite, which is equivalent to the matrix A being able to be expressed using finitely many infinite vectors. Or else it is infinite, which turns out to be equivalent to stating that the set of matrices that can be obtained from A by a finite number of row and column operations is Zariski-dense in the space of all infinite-by-infinite matrices.

For polynomial series, their strength fulfil a similar role. We define the strength of a polynomial series to be the infimum number of pairs of lower degree series needed to write down a formula for it. It is then true that the strength of a polynomial series is infinite if and only if the set of series obtained from it by finitely many substitutions is Zarisky-dense in the space it lives in. Both infinite-by-infinite matrices and polynomial series are examples of the following dichotomy: either you can express them using a finite amount of lower-dimensional data or their orbit under some group is dense.

This talk is about joint work with Jan Draisma, Rob Eggermont and Andrew Snowden that generalizes this statement to all finite-degree polynomial functors.