On the Classification of Exceptional Planar Functions over $\mathbb{F}_{p}$

TL;DR

We address the problem of classifying PN/planar monomial functions over $\mathbb{F}_p$ by focusing on the curve data of $f_t$ and the associated quotient $g_t=\frac{f_t}{x-y}$. Our approach combines the Weil bound, Bezout's theorem, and Bertini-type results to derive contradictions when $g_t$ would factor, after a careful singularity analysis of the projective curve $\chi_t$ and its intersection multiplicities. We establish multiple sufficient conditions guaranteeing the existence of an absolutely irreducible factor of $g_t$ over $\mathbb{F}_p$ in various parameter regimes, and separate the treatment into Case (A) and Case (B) to handle reducibility versus irreducibility. The results include extensive computational verification for many $t\le 1000$ with $p\in\{3,5,7\}$, providing substantial evidence toward Conjecture PN2 and PN3 and advancing the understanding of exceptional PN behavior in finite fields.

Abstract

We will present many strong partial results towards a classification of exceptional planar/PN monomial functions on finite fields. The techniques we use are the Weil bound, Bezout's theorem, and Bertini's theorem.

Theorems & Definitions (105)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Definition 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4