Littlestone and VC-dimension of families of zero sets

Abstract

We prove that, for any d linearly independent functions from some set into a d-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone dimension d − 1. Additionally, we characterize when such families are maximal of VC-dimension d − 1 and give a sufficient condition for when they are maximal of Littlestone dimension d − 1.