  Planar functions are special functions from a finite field to itself that
give rise to finite projective planes and other combinatorial objects. We
consider polynomials over a finite field $K$ that induce planar functions on
infinitely many extensions of $K$; we call such polynomials exceptional planar.
Exceptional planar monomials have been recently classified. In this paper we
establish a partial classification of exceptional planar polynomials. This
includes results for the classical planar functions on finite fields of odd
characteristic and for the recently proposed planar functions on finite fields
of characteristic two.
