Finding roots of a multivariate polynomial in a linear subspace

Suppose F is a polynomial of total degree d in t variables over a finite field k=Fqn. We are interested in finding roots of F that lie in a Fq-linear subspace of kt. For m≤n, we characterize a large class of m-dimensional Fq-subspaces U of kt such that the set of roots of F that lie in U can be bounded by dm in cardinality, independent of q, and constructed in expected time polynomial in n, t and dm.