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The Intersection Distribution: New Results and Perspectives

Sophie Huczynska, Lukas Klawuhn, Maura B. Paterson

TL;DR

The paper investigates the intersection distribution and non-hitting index for finite-field polynomials through dual algebraic and geometric lenses, tying the graph of a polynomial to the projective set $S_f\subset PG(2,q)$ and to $(q+1)$-arcs and ovals in finite geometry. It advances the theory by deriving new bounds and characterisations for the degree of $S_f$, exploring projective equivalence of polynomials, and providing both geometric and algebraic viewpoints for cubics and other families. It also furnishes examples of projectively inequivalent polynomials sharing the same intersection distribution, investigates the non-hitting spectrum beyond the smallest values, and presents two short proofs of the cubic case, thereby resolving several open questions posed by Li and Pott. The work highlights deep connections to algebraic curves, cyclotomy, and irreducible polynomials, with implications for Kakeya-type problems and related combinatorial structures, and paves the way for further study of equivalence notions and spectra in finite geometry.

Abstract

Intersection distribution and non-hitting index are concepts introduced recently by Li and Pott as a new way to view the behaviour of a collection of finite field polynomials. With both an algebraic interpretation via the intersection of a polynomial with a set of lines, and a geometric interpretation via a $(q+1)$-set possessing an internal nucleus, the concepts have proved their usefulness as a new way to view various long-standing problems, and have applications in areas such as Kakeya sets. In this paper, by exploiting connections with diverse areas including the theory of algebraic curves, cyclotomy and the enumeration of irreducible polynomials, we establish new results and resolve various Open Problems of Li and Pott. We prove geometric results which shed new light on the relationship between intersection distribution and projective equivalence of polynomials, and algebraic results which describe and characterise the degree of $S_f$ - the index of the largest non-zero entry in the intersection distribution of $f$. We provide new insights into the non-hitting spectrum, and show the limitations of the non-hitting index as a tool for characterisation. Finally, the benefits provided by the connections to other areas are evidenced in two short new proofs of the cubic case.

The Intersection Distribution: New Results and Perspectives

TL;DR

The paper investigates the intersection distribution and non-hitting index for finite-field polynomials through dual algebraic and geometric lenses, tying the graph of a polynomial to the projective set and to -arcs and ovals in finite geometry. It advances the theory by deriving new bounds and characterisations for the degree of , exploring projective equivalence of polynomials, and providing both geometric and algebraic viewpoints for cubics and other families. It also furnishes examples of projectively inequivalent polynomials sharing the same intersection distribution, investigates the non-hitting spectrum beyond the smallest values, and presents two short proofs of the cubic case, thereby resolving several open questions posed by Li and Pott. The work highlights deep connections to algebraic curves, cyclotomy, and irreducible polynomials, with implications for Kakeya-type problems and related combinatorial structures, and paves the way for further study of equivalence notions and spectra in finite geometry.

Abstract

Intersection distribution and non-hitting index are concepts introduced recently by Li and Pott as a new way to view the behaviour of a collection of finite field polynomials. With both an algebraic interpretation via the intersection of a polynomial with a set of lines, and a geometric interpretation via a -set possessing an internal nucleus, the concepts have proved their usefulness as a new way to view various long-standing problems, and have applications in areas such as Kakeya sets. In this paper, by exploiting connections with diverse areas including the theory of algebraic curves, cyclotomy and the enumeration of irreducible polynomials, we establish new results and resolve various Open Problems of Li and Pott. We prove geometric results which shed new light on the relationship between intersection distribution and projective equivalence of polynomials, and algebraic results which describe and characterise the degree of - the index of the largest non-zero entry in the intersection distribution of . We provide new insights into the non-hitting spectrum, and show the limitations of the non-hitting index as a tool for characterisation. Finally, the benefits provided by the connections to other areas are evidenced in two short new proofs of the cubic case.

Paper Structure

This paper contains 20 sections, 45 theorems, 56 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Geometric background
  3. Projective equivalence of polynomials
  4. Further examples of projectively inequivalent polynomials with the same intersection distribution
  5. Polynomials giving rise to the $(q+1)$-sets of Li and Pott's Examples 2.3 and 2.2
  6. Permutation Polynomials and their Inverses
  7. Algebraic viewpoint
  8. The degree of $S_f$
  9. Algebraic and geometric characterisations of $(q+1)$-sets
  10. Sets contained in the union of a small number of projective lines
  11. One projective line
  12. One projective line and one additional point
  13. Two projective lines
  14. Two lines and a point
  15. Two lines and two points
  16. ...and 5 more sections

Key Result

Theorem 1.5

Let $f$ be a polynomial over $\mathbb{F}_q$ and let $S_f$ be the associated $(q+1)$-set in ${\rm PG}(2,q)$. Then

Figures (2)

  • Figure 1: $(q+1)$-sets from (a) Example 2.3(1) and (b) Example 2.3(2) of LiPot
  • Figure 2: The first $(q+1)$-set of Example 2.3(2) of LiPot

Theorems & Definitions (102)

  • Definition 1.1: LiPot
  • Example 1.2: LiPot
  • Definition 1.3
  • Definition 1.4: LiPot
  • Theorem 1.5: LiPot
  • Definition 1.6
  • Theorem 1.7: LiPot
  • proof
  • Example 1.8
  • Theorem 1.9: TalScaf
  • ...and 92 more