A refinement of multivariate value set bounds

作者：Smith Luke^*; Wan Daqing

来源：Finite Fields and Their Applications, 2016, 38: 13-26.

DOI：10.1016/j.ffa.2015.11.002

• Value set
• Polynomial image set
• Multivariate polynomials
• Newton polytopes
• p-adic liftings

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 improvement of a degree matrix-based result given by Zan and Cao, making our new bound the strongest upper bound thus far.

• 出版日期2016-3