  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.
