This is a special case of [this question](https://mathoverflow.net/questions/262829/linear-combinations-of-low-degree-polynomials-with-agreement-guarantee).


Let $\mathbb{F}$ be a finite field and $\mathbb{F}\_{\leq d}[x,y]$ the set of bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$. Do there exist non-empty, disjoint subsets $A,B\subset\mathbb{F}\_{\leq d}[x,y]$ such that for all pairs $(\ell,P)$, where $\ell\subset\mathbb{F}^2$ is a line, and $P$ is a univariate polynomial of degree at most $d$ defined on $\ell$ we have: $$\Big|\{Q\in A:Q|\_\ell=P\}\Big|=\Big|\{Q\in B:Q|\_\ell=P\}\Big|.$$


**Follow-up Question:**
Do there exist $A,B\subset\mathbb{F}\_{\leq d}[x,y]$ satisfying the above and also $|A|=|B|=|\mathbb{F}|^{\mathcal{O}(1)}$?



Question about polynomials over finite fields
The condition that $A$ and $B$ are nonempty disjoint can be replaced with the condition that they are distinct, as we may just replace $A$ and $B$ with $A \cap (A - B)$ and $B \cap (A- B)$ respectively.


Let $q = |\mathbb F|$.


Let $A$ and $B$ be two random linear subspaces of the space of polynomials of degree $\leq d$ each of codimension $k$. As long as $0< k < { d+2 \choose 2}$, with high probability $A$ and $B$ are distinct. As long as $A$ and $B$ intersect transversely the space of polynomials vanishing on $\ell$, the number of elements of $A$ and $B$ taking a given fixed value on $\ell$ is $ q^{ {d+2 \choose 2} - k - (d+1)}$.


Equivalently, this happens when the perpendicular spaces of $A$ and $B$ intersect only at $0$ the space of linear forms on degree $d$ polynomaials that factor through reduction to $\ell$. Each nonzero linear form has a probability of $$\frac{(q^k-1) }{ q^{ {d+2 \choose 2}} -1}$$ of being in the perpendicular subspace, and there are at most $(q^{d+1}-1)$ nontrivial linear forms that factor through restriction to each of $q (q+1)$ lines, so as long as


$$ q(q+1) (q^{d+1}-1) \frac{(q^k-1) }{ q^{ {d+2 \choose 2}} -1}< 1$$


with high probability $A$ and $B$ are transverse to all these linear subspaces. This happens when $k<{d+2 \choose 2} - d-3$. So it looks to me like even random $d+4$-dimensional subspaces will do the job.


