More on the number of distinct values of a class of functions

Abstract

In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar functions over finite fields. This follows from a more general result proving that the upper bound cannot be tight for a much larger class of functions over an abelian group of order $y^n$ with $n>1$. Moreover, the tightness of the upper bound for the larger class of functions is equivalent to the existence of planar difference sets. To obtain better upper bounds, we first completely resolve an optimization problem involving the partitioning of a number into triangular parts. Our solution, which is algorithmic and constructive, allows us to determine tight upper bounds provided the relevant parameters are given explicitly. We also provide a suite of upper bounds which can be applied across a range of parameters. These are established via a well-studied Diophantine equation and are related to class numbers of quadratic number fields.