Computing implicitizations of multi-graded polynomial maps

In this paper, we focus on computing the kernel of a map of polynomial rings. This core problem in symbolic computation is known as implicitization. While Gröbner basis methods can be used to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the polynomial map is multigraded, we consider an alternative approach. We first demonstrate how to quickly compute a matrix of maximal rank for which a polynomial map has a positive multigrading. We then describe how minimal generators in each graded component of the kernel can be computed with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where standard techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is embarassingly parallel. A fully parallelized version of our algorithm is in development in both Macaulay2 and OSCAR.