# Rational points on certain quintic hypersurfaces - Mathematics > Number Theory

Abstract: Let $fx=x^5+ax^3+bx^2+cx \in \Zx$ and consider the hypersurface of degreefive given by the equation \cal{V} {f}: fp+fq=fr+fs. Under theassumption $b eq 0$ we show that there exists $\Q$-unirational ellipticsurface contained in $\cal{V} {f}$. If $b=0, a<0$ and $-a ot\equiv 2,18,34\pmod {48}$ then there exists $\Q$-rational surface contained in $\cal{V} {f}$.Moreover, we prove that for each $f$ of degree five there exists$\Qi$-rational surface contained in $\cal{V} {f}$.

Author: Maciej Ulas

Source: https://arxiv.org/