Sheet3 Q3

Let $P(x, y, z)$ be a homogeneous polynomial of degree $d$ defining a nonsingular curve $C$ in $ℂℙ^2$. (i) Write down Euler's relation for $P, P_x, P_y, P_z$. Deduce that the Hessian determinant satisfies: \[ z ℋ_P(x, y, z)=(d-1) \det\begin{pmatrix} P_{xx} & P_{xy} & P_{xz} \\ P_{yx} & P_{yy} & P_{yz} \\ P_x& P_y & P_z \end{pmatrix} \] Solution. Euler's relation $dP=xP_x+yP_y+zP_z$. Both sides $∂_x$ we get $dP_x=P_x+xP_{xx}+yP_{yx}+zP_{zx}$, so \[(d-1)P_x=xP_{xx}+yP_{yx}+zP_{zx}\] Both sides $∂_y$ we get $dP_y=P_y+xP_{xy}+yP_{yy}+zP_{zy}$, so \[(d-1)P_y=xP_{xy}+yP_{yy}+zP_{zy}\] Both sides $∂_z$ we get $dP_z=P_z+xP_{xz}+yP_{yz}+zP_{zz}$, so \[(d-1)P_z=xP_{xz}+yP_{yz}+zP_{zz}\] By definition of Hessian and linearity of det, \[ z ℋ_P(x, y, z)=\det\begin{pmatrix} P_{xx} & P_{xy} & P_{xz} \\ P_{yx} & P_{yy} & P_{yz} \\ zP_{zx} &zP_{zy}&zP_{zz} \end{pmatrix}\] Adding $x$ times first row, $y$ times second row to third row \[=(d-1) \det\begin{pmatrix} P_{xx} & P_{xy} & P_{xz} \\ P_{yx} & P_{yy} & P_{yz} \\ P_x& P_y & P_z \end{pmatrix}. ∎ \] (ii) Deduce further that: \[ z^2 ℋ_P(x, y, z)=(d-1)^2 \det\begin{pmatrix} P_{xx} & P_{xy} & P_x\\ P_{yx} & P_{yy} & P_y \\ P_x& P_y & d P /(d-1) \end{pmatrix} \] Solution. Adding $x$ times first column, $y$ times second column to third column, we get the determinant. ∎ (iii) Deduce that if $P(x, y, 1)=y-g(x)$ then $[a, b, 1]$ is a point of inflection of $C$ if and only if $b=g(a)$ and $g''(a)=0$. This shows the lectures definition of points of inflection corresponds to the usual notion of a point of inflection of the graph of a function $g(x)$ on $ℝ$ or $ℂ$. Solution. point$∈C⇔P(a,b,1)=0⇔b=g(a)$. By (ii), \[ z^2 ℋ_P(x, y, z)=(d-1)^2 \det\begin{pmatrix} -g''(x)&0&-g'(x)\\ 0&0&1\\ -g'(x)&1&0 \end{pmatrix}=(d-1)^2g''(x) \] so $ℋ_P(a,b,1)=0⇔g''(a)=0$ (if $d=1$, $g''=0$).