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| -rw-r--r-- | bezout.tex | 4 |
1 files changed, 2 insertions, 2 deletions
@@ -48,7 +48,7 @@ $\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be interpreted as a real function of $2N$ real variables. The number of critical points it has is given by the usual Kac--Rice formula: \begin{equation} \label{eq:real.kac-rice} - \mathcal N(\epsilon) + \mathcal N(\kappa,\epsilon) = \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \left|\det\begin{bmatrix} \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\ @@ -66,7 +66,7 @@ determinant that appears above is equivalent to $|\det\partial\partial H|^2$. This allows us to write the \eqref{eq:real.kac-rice} in the manifestly complex form \begin{equation} \label{eq:complex.kac-rice} - \mathcal N(\epsilon) + \mathcal N(\kappa,\epsilon) = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) |\det\partial\partial H|^2. \end{equation} |