Merge branch 'master' of https://git.overleaf.com/5fcce4736e7f601ffb7e1484

1 files changed, 11 insertions, 2 deletions

diff --git a/bezout.tex b/bezout.tex
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--- a/bezout.tex
+++ b/bezout.tex
@@ -23,12 +23,21 @@
 \date\today
 
 \begin{abstract}
-  We study the saddle-points of the $p$-spin model, the best understood example of `complex (rugged) landscape' in the space in which
-  saturates the Bézout bound \cite{Bezout_1779_Theorie}.
+  We study the saddle-points of the $p$-spin model, the best understood example of `complex (rugged) landscape' in the space in which all its $N$ variables are allowed to be complex. The problem becomes
+  a system of $N$ random equations of degree $p-1$. 
+  We solve for quantities averaged over randomness in the $N \rightarrow \infty$ limit. 
+  We show that the number of solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1) $\cite{Bezout_1779_Theorie}.
+The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b B^2 +ic [A,B]_-+ d [A,B]_+$,
+where $A$ and $B$ are GOE matrices and $a-d$ real. Its spectrum  has a transition 
+from one-cut to two-cut that generalizes the notion  of `threshold level' that is well-known in the real problem.
+In the case that the disorder is itself real, only the square-root of the total  number solutions are real.
+In terms of real and imaginary parts of the energy, the solutions are divided in  sectors where the saddles have
+different topological properties.
 \end{abstract}
 
 \maketitle
 
+
 \begin{equation} \label{eq:bare.hamiltonian}
   H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}