# The algebraic matroid of the funtf variety

## TL;DR

This paper studies the algebraic structure of finite unit norm tight frames using matroid theory, characterizing bases in three dimensions and providing bounds for higher dimensions.

## Contribution

It characterizes the algebraic matroid bases of finite unit norm tight frame varieties in three dimensions and offers partial results and combinatorial bounds for higher dimensions.

## Key findings

- Characterization of matroid bases in $\,\mathbb{R}^3$.
- Partial results for $\,\mathbb{R}^n$, $n\ge 4$.
- Method to bound projection degrees using combinatorics.

## Abstract

A finite unit norm tight frame is a collection of $r$ vectors in $\mathbb{R}^n$ that generalizes the notion of orthonormal bases. The affine finite unit norm tight frame variety is the Zariski closure of the set of finite unit norm tight frames. Determining the fiber of a projection of this variety onto a set of coordinates is called the algebraic finite unit norm tight frame completion problem. Our techniques involve the algebraic matroid of an algebraic variety, which encodes the dimensions of fibers of coordinate projections. This work characterizes the bases of the algebraic matroid underlying the variety of finite unit norm tight frames in $\mathbb{R}^3$. Partial results towards similar characterizations for finite unit norm tight frames in $\mathbb{R}^n$ with $n \ge 4$ are also given. We provide a method to bound the degree of the projections based off of combinatorial~data.