Counterexamples to Dembowski and Ostrom conjecture on Planar function

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 $p\ge 3$ that are not of Dembowski-Ostrom type.

There is some error. It gives some results but not disproving the conjecture