Odd-dimensional generalized finite-field distance-set conjecture

Let $q$ be a prime power of sufficiently large characteristic and let

$$P(x)=\sum_{j=1}^d a_jx_j^c\in\mathbb F_q[x_1,\dots,x_d],$$

where $a_j\ne 0$ and $c\geq 2$. For subsets $E,F\subset\mathbb F_q^d$ and odd $d\geq 3$, define

$$\Delta_P(E,F)=\{P(x-y):x\in E,\ y\in F\}.$$

Generalized finite-field distance conjecture. One has

$$|\Delta_P(E,F)|\gtrsim \min\left(q,q^{-\frac{d-1}{2}}\sqrt{|E||F|}\right).$$

This conjecture seeks a sharp generalized distance-set lower bound in odd dimensions, extending quadratic finite-field distance estimates to diagonal polynomial forms. The source introduces it as a conjecture and explains that the odd-dimensional case is the focus; no resolution is supplied.