On planar functions over ${\mathbb{F}}_{q^3}$

Let ${\mathbb{F}}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over ${\mathbb{F}}_{q^3}$ of the family $$f_{E,A,B,C,D}(X):=E{X}^2+A{X}^{q+1}+B{X}^{q^2+1}+C{X}^{2q}+D{X}^{2q^2}\in {\mathbb{F}}_q[X]\mathrm{.}$$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.