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| -rw-r--r-- | bezout.tex | 18 |
1 files changed, 12 insertions, 6 deletions
@@ -217,7 +217,10 @@ where \end{aligned} \end{equation} \end{widetext} -This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes as $a\to1$, where we should recover known results for the real $p$-spin. +This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot +z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter +vanishes as $a\to1$, where we should recover known results for the real +$p$-spin. \begin{figure}[htpb] \centering @@ -255,7 +258,8 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of \end{equation} which makes its ensemble that of Gaussian complex symmetric matrices. Given its variances $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and -$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse +$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is +constant inside the ellipse \begin{equation} \label{eq:ellipse} \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+ \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2 @@ -342,7 +346,8 @@ gives =(p-1)^N. \end{equation} This is precisely the Bézout bound, the maximum number of solutions that $N$ -equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is gained by looking at the count as a function of $a$, defined by +equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is +gained by looking at the count as a function of $a$, defined by \begin{equation} \label{eq:count.def.marginal} \overline{\mathcal N}(\kappa,\epsilon) =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a) @@ -376,9 +381,10 @@ contained within is =(p-1)^{N/2}, \end{equation} the square root of \eqref{eq:bezout} and precisely the number of critical -points of the real pure spherical $p$-spin model. (note the role of conjugation symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In fact, the full -$\epsilon$-dependence of the real pure spherical $p$-spin is recovered by this -limit as $\epsilon$ is varied. +points of the real pure spherical $p$-spin model. (note the role of conjugation +symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In +fact, the full $\epsilon$-dependence of the real pure spherical $p$-spin is +recovered by this limit as $\epsilon$ is varied. \begin{figure}[htpb] \centering |