Rational Points on Certain Quintic Hypersurfaces

Let f(x) = x+ax+bx+cx ∈ Z[x] and consider the hypersurface of degree five given by the equation Vf : f(p) + f(q) = f(r) + f(s). Under the assumption b 6= 0 we show that there exists Q-unirational elliptic surface contained in Vf . If b = 0, a < 0 and −a 6≡ 2, 18, 34 (mod 48) then there exists Q-rational surface contained in Vf . Moreover, we prove that for each f of degree five there exists Q(i)-rational surface contained in Vf .