Hidden field equations - existence of zeroes

Let $\mathbb{F}_q$ be a finite field of size $q$ (prime), and $\mathbb{F}_{q^n}$ be a degree-$n$ algebraic extension of $\mathbb{F}_q$.

Let $F$ be a polynomial function $\mathbb{F}_{q^n} \to \mathbb{F}_{q^n}$ of the form $$ \sum_{i, j \in I_A} A_{i,j} X^{q^i + q^j} + \sum_{i\in I_B} B_i X^{q^i} + C $$ where $A_{i,j}, B_i,$ and $C$ are some constants in $\mathbb{F}_{q^n}$.

Given a random $D \in \mathbb{F}_{q^n}$, we need to find a solution $X$ for $F(X) = D$.

My question is: why does such a solution exist? Does the range of $F$ cover $\mathbb{F}_\mathbb{q^n}$? How do we check?

Such a solution does not necessarily exist unless $F(X)$ is a permutation polynomial over $\mathbb F_{q^n}$.

Permutation polynomials are precisely those whose range covers the whole field.

The paper Recognising permutation functions in polynomial time (by Neeraj Kayal, one of the authors of the AKS polynomial time primality test) gives a polynomial time test.