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\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 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
+ example of a `complex' (rugged) landscape -- when its $N$ variables are
+ complex. These points are the solutions to a system of $N$ random equations
+ of degree $p-1$. We solve for $\overline{\mathcal{N}}$, the number of
+ solutions averaged over randomness in the $N\to\infty$ limit. We find that
+ it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$.
+ 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. The
+ results from the real problem are recovered in the limit of real disorder. In
+ this case, 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}