The following problem is motivated by this MO question on rich directions determined by a set of a finite plane.
Problem Does there exist a constant $C$ such that for all odd primes $p$ there is a polynomial $f(x)=x^{2p}+g(x)x^p+h(x)\in\mathbb F_p[x]$ satisfying
- $deg(g)<\tfrac p 3 +C$;
- $deg(h)<\tfrac {2p} 3 +C$;
- $f(x)$ is fully reducible (it factors completely, as a product linear factors, over $\mathbb F_p$);
- $f(x)$ is not of the form $f(x)=(x^p-x)^2$ or $f(x)=(x-t)^p(x-s)^p$ or $f(x)=(x^p-x)(x-t)^p$?
Here is an example of a similarly lacunary fully reducible polynomial, but satisfying $deg(g),deg(h)<\tfrac {p} 2 +C$ instead: $$f(x)=(x^{(p+1)/2}-x)^2(x^{p-1}-1).$$
I also add an approach that does not work. One might search for a polynomial of the form $f(x)=(x^p+a(x))(x^p+b(x))$ with $deg(a),deg(b)<\tfrac p 3 +C$. Unfortunately by a result of Rédei such polynomials are fully reducible only if $a(x),b(x)$ are constant or equal to $-x$.
