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
:$\left, \begin x&x'&x''&x\\ x_1&x_1'&x_1''&x_1\\ x_2&x_2'&x_2''&x_2\\ x_3&x_3'&x_3''&x_3\\ \end\ = 0.$
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 $t\to f(t), x\to g(t)^x$, 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 is that the cubic differential is invariant under the automorphism group $PGL(2,\mathbf R)$ of the projective line. More precisely, it is invariant under $t\to\frac$, $dt\to\fracdt$, and $x\to C(ct+d)^x$.

The invariant vanishes identically if (and only if) the curve is a conic section. Points where 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.

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Projective geometry