# Computing symmetric determinantal representations

## TL;DR

This paper presents a Macaulay2 package that efficiently computes definite symmetric determinantal representations for real polynomials, focusing on quadrics and low-degree plane curves, using linear algebra and numerical algebraic geometry.

## Contribution

Introduction of a new Macaulay2 package that computes symmetric determinantal representations without genericity assumptions, emphasizing speed and robustness.

## Key findings

- Successfully computes representations for quadrics and low-degree curves.
- Employs linear algebra and numerical methods for robustness.
- No assumptions on polynomial genericity.

## Abstract

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our algorithms are geared towards speed and robustness, employing linear algebra and numerical algebraic geometry, without genericity assumptions on the polynomials.