Solutions in projective space

We show how to solve system defined in projective space..

Solving systems defined in projective space

In the guides so far we were computing solutions in $\mathbb{R}^n$ or $\mathbb{C}^n$. In some applications, however, it is required to compute solutions in projective space $\mathbb{RP}^n$ or $\mathbb{CP}^n$ . This space is defined as the space of all lines in $\mathbb{R}^{n+1}$, respectively $\mathbb{C}^{n+1}$, passing through the origin. HomotopyContinuation.jl automatically recognizes systems defined over projective space and adjusts the output. Next, we show an example.

Example: computing the degree of a projective variety

Consider the projective variety in the 2-dimensional complex projective space $\mathbb{CP}^2$. $V = \{ x^2 + y^2 - z^2 = 0 \}$

The degree of $V$ is the number of intersection points of $V$ with a generic line. Let us see what it is. First we initialize the defining equation of $V$.

using HomotopyContinuation
@polyvar x y z
V = x^2 + y^2 - z^2;

Let us sample the equation of a random line $L$.

L = randn(1,3) * [x, y, z];

Now we compute the number of solutions to $[V, L]=0$.

solve([V; L])
ProjectiveResult with 2 tracked paths
==================================
• 2 non-singular solutions (2 real)
• 0 singular finite solutions (0 real)
• 0 failed paths
• random seed: 534036

We find two distinct solutions and conclude that the degree of $V$ is 2. In particular, the output does not show solutions at infinity, simply because this concept is not defined in projective space.