# Examples

Polynomial equations appear in a wide variety of topics.

## Applications in the Sciences

#### Polynomial systems arising from Nash equilibria

A story about how game theory meets multilinear systems and product of simplices.

#### Alt's problem

A classical problem from robotics

#### Computer vision

Triangulation of a dinosaur

#### Bacillus Subtilis

environmental and energy stress response of the bacterium Bacillus subtilis

#### Cyclooctane

The conformation space of cyclooctane

#### The 6R-serial link robot

Using HomotopyContinuation.jl for a problem in kinematics

#### Energy minimization

Computing a macroscopic configuration

## Mathematics of Data

#### The reach of a plane curve

Computing the reach

#### Sampling and Homology via Bottlenecks

Sampling an algebraic variety and computing its homology

#### Computing a logarithmic Voronoi cell

An adventure at the intersection of statistics and nonlinear algebra.

#### Sampling a probability distribution on a variety

By intersecting with linear spaces

#### Method of moments

For mixtures of three Gaussians

#### The point on a variety that minimizes the distance to a given point

Computing critical points

#### Monte Carlo integration

Integrating functions on a variety

## Algebraic Geometry

#### When do two ellipses meet?

Let two ellipses grow until they intersect

#### Symmetroids

Computing the degree of a cubic symmetroid

#### Tensor decomposition and homotopy continuation

Using homotopy continuation for computing the CPD of a tensor

#### Tritangents

Using HomotopyContinuation.jl for computing tritangents of sextic curves

#### Curvature of plane curves

How to compute curvature

#### 3264 conics tangent to five conics

All of them can be real.

#### The number of circles that are tangent to 3 given conics

Using HomotopyContinuation.jl for a problem in enumerative geometry