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Classification of a class of planar quadrinomials

Chin Hei Chan, Maosheng Xiong

TL;DR

This work classifies planar DO quadrinomials of the form $f_{oldsymbol{c}}(X)$ over $\mathbb{F}_{q^2}$ for odd $q$, by executing a geometric analysis of the accompanying rational function $g_{oldsymbol{c}}(X)=\dfrac{B_{oldsymbol{c}}(X)}{A_{oldsymbol{c}}(X)}$ and its ramification structure. Building on the method of Ding and Zieve, the authors derive seven linear equivalence classes for $g(X)$ and translate these into seven linear classes for $f(X)$; among these, only three yield planar functions: $X^{Q+1}$ (when $\frac{\ell}{\gcd(k,\ell)}$ is even), $X^{Q+q}$ (when $\frac{k\ell}{\gcd(k,\ell)^2}$ is odd), and $P_2(x,y)=(x^Qy, x^{Q+1}+\varepsilon y^{Q+1})$ with $\frac{k}{\gcd(k,\ell)}$ odd and $\varepsilon$ a non-square in $\mathbb{F}_q^*$; additionally, if $k\mid \ell$, the planar case reduces to $X^2$. The paper also treats Cases 6 and 7 as non-planar via Appendix arguments, underscoring the completeness of the classification. The results reinforce the equivalence of linear/EA/CCZ notions for these DO planar functions and provide a framework for applying algebraic-geometry techniques to odd-characteristic planar problems with DO structure.

Abstract

Let $p$ be an odd prime, $k,\ell$ be positive integers, $q=p^k, Q=p^{\ell}$. In this paper we characterise planar functions of the form $f_{\underline{c}}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\mathbb{F}_{q^2}$ for any $\underline{c}=(c_0,c_1,c_2,c_3) \in \mathbb{F}_{q^2}^4$ in terms of linear equivalence.

Classification of a class of planar quadrinomials

TL;DR

This work classifies planar DO quadrinomials of the form over for odd , by executing a geometric analysis of the accompanying rational function and its ramification structure. Building on the method of Ding and Zieve, the authors derive seven linear equivalence classes for and translate these into seven linear classes for ; among these, only three yield planar functions: (when is even), (when is odd), and with odd and a non-square in ; additionally, if , the planar case reduces to . The paper also treats Cases 6 and 7 as non-planar via Appendix arguments, underscoring the completeness of the classification. The results reinforce the equivalence of linear/EA/CCZ notions for these DO planar functions and provide a framework for applying algebraic-geometry techniques to odd-characteristic planar problems with DO structure.

Abstract

Let be an odd prime, be positive integers, . In this paper we characterise planar functions of the form over for any in terms of linear equivalence.
Paper Structure (19 sections, 19 theorems, 130 equations)

This paper contains 19 sections, 19 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Statement of main results
  4. Preliminaries
  5. Self-conjugate reciprocal polynomials
  6. Rational Functions
  7. Branch points and ramification
  8. Linear equivalence, EA-equivalence and CCZ-equivalence
  9. Auxiliary lemmas
  10. Geometric properties of $g(X)$
  11. Classification of $g(X)$
  12. Linear Equivalence Classes of $f(X)$
  13. Proof of Theorem \ref{['Main']}
  14. Cases 1)-5)
  15. Case $k \mid \ell$
  16. ...and 4 more sections

Key Result

Theorem 1

Let $p$ be an odd prime, $k$ and $\ell$ positive integers, $q=p^k$ and $Q=p^\ell$. Consider the quadrinomial $f_{\underline{c}}(X)$ given in (DO) for any $\underline{c}=(c_0,c_1,c_2,c_3) \in \mathbb{F}_{q^2}^4$. Then $f_{\underline{c}}(X)$ is planar over $\mathbb{F}_{q^2}$ if and only if it is linea Moreover, if $k \mid \ell$, then $f_{\underline{c}}(X)$ is planar if and only if it is linear equiv

Theorems & Definitions (30)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Definition 1
  • Definition 2
  • ...and 20 more