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

Robert Coulter, Steven Senger

TL;DR

The paper proves that the previously known upper bound on the image size $V(f)$ for several function classes, including planar functions, cannot be tight over finite fields. It connects tightness to planar difference sets and projective planes, and proceeds to develop an algorithmic framework based on a cost function and rhombic floor to obtain explicit, often tight, upper bounds for $V(f)$ across general and special cases of $q$. Central to the method are reductions tied to triangular (and rhombic) decompositions, Lebesgue–Ramanujan–Nagell type Diophantine equations, and class-number obstructions, culminating in refined bounds for square and non-square prime powers via the modified class divisor concept. The results yield both general bounds and sharp bounds in numerous cases, with an extensive appendix detailing the exceptional configurations where greedy cost-minimization fails and how to realize the minimal costs. Together, these contributions advance understanding of how far planar and related functions are from being permutations and illuminate the deep interplay between finite geometry, Diophantine equations, and number-theoretic invariants in bounding image sizes.

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.

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

TL;DR

The paper proves that the previously known upper bound on the image size for several function classes, including planar functions, cannot be tight over finite fields. It connects tightness to planar difference sets and projective planes, and proceeds to develop an algorithmic framework based on a cost function and rhombic floor to obtain explicit, often tight, upper bounds for across general and special cases of . Central to the method are reductions tied to triangular (and rhombic) decompositions, Lebesgue–Ramanujan–Nagell type Diophantine equations, and class-number obstructions, culminating in refined bounds for square and non-square prime powers via the modified class divisor concept. The results yield both general bounds and sharp bounds in numerous cases, with an extensive appendix detailing the exceptional configurations where greedy cost-minimization fails and how to realize the minimal costs. Together, these contributions advance understanding of how far planar and related functions are from being permutations and illuminate the deep interplay between finite geometry, Diophantine equations, and number-theoretic invariants in bounding image sizes.

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 with . 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.
Paper Structure (14 sections, 23 theorems, 107 equations, 1 figure)

This paper contains 14 sections, 23 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction and history
  2. Motivation
  3. Correspondence between the upper bound and a projective plane of order $t-1$
  4. Showing the upper bound is nigh impossible
  5. First step towards improving the upper bound
  6. An algorithmic determination of $\mathop{\mathrm{cost}}\nolimits(n)$
  7. The general situation
  8. The case where $q$ is a square
  9. More on the Lebesgue-Ramanujan-Nagell equation
  10. The case where $q$ is not a square
  11. The case where $q=p^n$ is not a square
  12. The Modified Class Divisor
  13. A new bound for $V(f)$ when $q=p^n$ is not a square
  14. Appendix: Costs of the exceptions

Key Result

Theorem 1

The following statements hold:

Figures (1)

  • Figure 1: Here we see the circle $C_n$ centered at $\left(\frac{1}{2},\frac{1}{2}\right)$, and $L_{\alpha_0}$ intersecting $C_n$ at $(x_0,y_0)$ and $(y_0,x_0).$ Also pictured are examples of lines $L_\alpha$ with $\alpha>\alpha_0,$ and $L_\beta$ with $\beta<\alpha_0.$

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Corollary 7
  • Lemma 8
  • ...and 27 more