Rational solutions of pairs of diagonal equations, one cubic and one quadratic

TL;DR

This paper proves that pairs of diagonal equations, one cubic and one quadratic, have non-trivial integer solutions when the number of variables exceeds 10, based on new estimates for exponential sums.

Contribution

It provides an essentially optimal estimate for a specific exponential sum and establishes the existence of solutions under certain variable count conditions.

Findings

Non-trivial solutions exist for pairs of diagonal cubic and quadratic equations with over 10 variables.

An optimal estimate for the 32/3 moment of a particular exponential sum was derived.

The results depend on modest local solubility assumptions.

Abstract

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $\alpha x^3+\beta x^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10.