Colloquium Mathematics - Dr. M.E. Hochstenbach

Title:

Solving polynomial systems via determinantal representations" (joint work with Bor Plestenjak, Ada Boralevi, Jan Draisma, and Jasper van Doornmalen)

Abstract:

We consider a systems of 2 equations of polynomials of 2 variables: p(x,y) = 0, q(x,y) = 0. Roots of p(x) = 0, a polynomial of 1 variable, are usually solved via a companion matrix approach: a matrix A such that det(A-xI) = p(x). For a polynomial of 2 variables the challenge is to find a determinantal representation: matrices A, B, and C such that det(A-xB-yC) = p(x,y), and similar for q. If the matrices are small enough, this construction enables us to efficiently solve for the roots. This fascinating topic includes open questions of over a century old.