Representation of Squares by Nonsingular Cubic Forms

TL;DR

This paper establishes an asymptotic count for representing squares using nonsingular cubic forms in six or more variables, employing advanced exponential sum estimates and the Circle Method.

Contribution

It introduces new exponential sum bounds and extends the Circle Method to analyze representations of squares by cubic forms in six or more variables.

Findings

Asymptotic formula for representations of squares

Extension of exponential sum bounds to cubic forms

Application of Heath-Brown's Circle Method

Abstract

We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown's form of the Circle Method and various exponential sum results. The depth of the exponential sum results is comparable to Hooley's work on cubic forms in nine variables, in particular we prove an analogue of Katz' bound.