  Over finite fields, if the image of a polynomial map is not the entire field,
then its cardinality can be bounded above by a significantly smaller value.
Earlier results bound the cardinality of the value set using the degree of the
polynomial, but more recent results make use of the powers of all monomials.
  In this paper, we explore the geometric properties of the Newton polytope and
show how they allow for tighter upper bounds on the cardinality of the
multivariate value set. We then explore a method which allows for even stronger
upper bounds, regardless of whether one uses the multivariate degree or the
Newton polytope to bound the value set. Effectively, this provides an alternate
proof of Kosters' degree bound, an improved Newton polytope-based bound, and an
improvement of a degree matrix-based result given by Zan and Cao.
