Counterexamples to Dembowski and Ostrom conjecture on Planar function

TL;DR

This paper constructs specific classes of planar polynomials over finite fields of characteristic p≥3 that serve as counterexamples to the Dembowski-Ostrom conjecture, which posited all planar polynomials are of a particular form.

Contribution

The paper provides explicit counterexamples of planar polynomials not of Dembowski-Ostrom type, challenging a longstanding conjecture in finite field theory.

Findings

Counterexamples to the Dembowski-Ostrom conjecture are constructed.

Planar polynomials outside the Dembowski-Ostrom class exist over any finite field with characteristic p≥3.

Abstract

Let $F_{q}$ be a finite field of cardinality $q$. A polynomial over finite field $F_{q}$ of the form $\sum_{i,j}a_{ij}x^{p^{i}+p^{j}}$ is called a Dembowski-Ostrom (DO) polynomial. The Dembowski-Ostrom conjecture says that a planar polynomial is necessarily a Dembowski-Ostrom polynomial. In this article, we construct certain classes of planar polynomials over any finite field of characteristic $p\geq 3$ that are not of Dembowski-Ostrom type.