On planar functions of elementary abelian $p$-group type

Abstract

We proved affine planes corresponding to quadratic planar functions over $F_{p^n}$ are semifield planes, and we determined affine planes corresponding to planar functions $f(x) = x^10 − αx^6 − α^2x^2$ by Ding and Yuan. Moreover we calculated explicit shapes of planar functions from the square mappings of almost all known finite commutative semifields.