let \mathbb{F} be a finite field of size n, and let P be the set of all planes in \mathbb{F}^3 (meaning affine subspaces of dimension 2).
S \subset P; \; \; |S|\geq\frac{2(d+1)}{n} |P|; \; \; 100d<n. also A is a function that assigns to each s in S a degree d polynomial over s, such that if s, s' are two non parallel planes in S then A(s)=A(s') on the intersection s \cap s'.

I need to show that there is a polynomial Q of degree 2d on \mathbb{F}^3 such that Q\mid_{s}=A(s) for each s\in S

what I tried to do is take d parallel planes in S (which I can find using the pigeonhole principle) and use interpolation over them to find Q. now, for each plane s the isn't parallel to them, it intersect them in d*n points. if deg(Q\mid_s)=d then I could use the Shwartz Zippel theorem to show that Q\mid_s = A(s), but all I know is that the degree is at most 2d, and not d.

another idea, is to find 2d parallel plane, and to interpolate over d of them, and somehow show that the polynomial agrees on the other planes as well, but I don't know how to do that. the Q polynomial from above is d degree over all plane parallel to the planes that were in the interpolation, so maybe this could help