# Linearly dependent powers of binary quadratic forms

## TL;DR

This paper investigates the minimal number of binary quadratic forms whose powers are linearly dependent, establishing bounds and providing explicit examples for various degrees.

## Contribution

It determines bounds on the number of forms for linear dependence of their powers and constructs explicit examples for degrees greater than or equal to four.

## Key findings

- For r ≤ 4, the degree d ≤ 5.
- For d ≥ 4, constructed sets with r = ⌊d/2⌋ + 2.
- Provided explicit examples and techniques for generating such forms.

## Abstract

Given an integer $d \ge 2$, what is the least $r$ so that there is a set of binary quadratic forms $\{f_1,\dots,f_r\}$ for which $\{f_j^d\}$ is non-trivially linearly dependent? We show that if $r \le 4$, then $d \le 5$, and for $d \ge 4$, construct such a set with $r = \lfloor d/2\rfloor + 2$. Many explicit examples are given, along with techniques for producing others.