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 ${\displaystyle p:\mathbf {P} ^{1}\to \mathbf {P} ^{2}}$ is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by ${\displaystyle p(t)=(x_{1}(t),x_{2}(t),x_{3}(t))}$ then associated to p is the third-order ordinary differential equation

${\displaystyle \left|{\begin{matrix}x&x'&x''&x'''\\x_{1}&x_{1}'&x_{1}''&x_{1}'''\\x_{2}&x_{2}'&x_{2}''&x_{2}'''\\x_{3}&x_{3}'&x_{3}''&x_{3}'''\\\end{matrix}}\right|=0.}$

Generically, this equation can be put into the form

${\displaystyle x'''+Ax''+Bx'+Cx=0}$

where ${\displaystyle A,B,C}$ are rational functions of the components of p and its derivatives. After a change of variables of the form ${\displaystyle t\to f(t),x\to g(t)^{-1}x}$, this equation can be further reduced to an equation without first or second derivative terms

${\displaystyle x'''+Rx=0.}$

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

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

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 Thorbergsson & Umehara (2002), depending on the curve's homotopy class in the projective plane.