Laguerre–Forsyth invariant

In projective geometry, the Laguerre–Forsyth invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew Forsyth, the latter of whom analyzed the invariant in an influential book on ordinary differential equations.

Suppose that $$p:\mathbf P^1\to\mathbf P^2$$ is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by $$p(t)=(x_1(t),x_2(t),x_3(t))$$ then associated to p is the third-order ordinary differential equation
 * $$\left|\begin{matrix}

x&x'&x&x'\\ x_1&x_1'&x_1&x_1'\\ x_2&x_2'&x_2&x_2'\\ x_3&x_3'&x_3&x_3'\\ \end{matrix}\right| = 0.$$ Generically, this equation can be put into the form
 * $$x'+Ax+Bx'+Cx = 0$$

where $$A,B,C$$ are rational functions of the components of p and its derivatives. After a change of variables of the form $$t\to f(t), x\to g(t)^{-1}x$$, this equation can be further reduced to an equation without first or second derivative terms
 * $$x''' + Rx = 0.$$

The invariant $$P=(f')^2R$$ is the Laguerre–Forsyth invariant.

A key property of $P$ is that the cubic differential $P(dt)^{3}$ is invariant under the automorphism group $$PGL(2,\mathbf R)$$ of the projective line. More precisely, it is invariant under $$t\to\frac{at+b}{ct+d}$$, $$dt\to\frac{ad-bc}{(ct+d)^2}dt$$, and $$x\to C(ct+d)^{-2}x$$.

The invariant $P$ vanishes identically if (and only if) the curve is a conic section. Points where $P$ vanishes are called the sextactic points of the curve. It is a theorem of Herglotz and Radon that every closed strictly convex curve has at least six sextactic points. This result has been extended to a variety of optimal minima for simple closed (but not necessarily convex) curves by, depending on the curve's homotopy class in the projective plane.