Rational points on certain quintic hypersurfaces

TL;DR

This paper investigates rational points on specific quintic hypersurfaces defined by a polynomial equation, establishing the existence of rational surfaces under various conditions and showing the presence of -rational surfaces for all such polynomials.

Contribution

It demonstrates the existence of -unirational elliptic surfaces and -rational surfaces within these quintic hypersurfaces under certain coefficient conditions, and proves the universal presence of (i)-rational surfaces.

Findings

Existence of -unirational elliptic surface when b .

Existence of -rational surface when b=0, a<0, and certain congruence conditions.

Presence of (i)-rational surface for all degree five polynomials.

Abstract

Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic surface contained in $\cal{V}_{f}$. If $b=0, a<0$ and $-a\not\equiv 2,18,34 \pmod {48}$ then there exists $\Q$-rational surface contained in $\cal{V}_{f}$. Moreover, we prove that for each $f$ of degree five there exists $\Q(i)$-rational surface contained in $\cal{V}_{f}$.