diff options
| -rw-r--r-- | bezout.tex | 4 |
1 files changed, 2 insertions, 2 deletions
@@ -336,7 +336,7 @@ values of $\kappa$ and $\epsilon$. Taking this saddle gives \log\overline{\mathcal N}(\kappa,\epsilon) =N\log(p-1). \end{equation} -This is precisely the Bézout bound, the maximum number of solutions that $N$ +This is, to this order, 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 $\overline{\mathcal N}(\kappa,\epsilon,a)=e^{Nf(a)}$. In the large-$N$ limit, this is the @@ -364,7 +364,7 @@ Notice that the limit of this expression as $a\to\infty$ corresponds with several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to negative infinity as $a\to1$, i.e., as the spins are restricted to the reals. This is natural, given that the $y$ contribution to -the volume shrinks to zero as that of an $N$-dimensional sphere with volume +the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(a-1)$ with volume $\sim(a-1)^N$. However, when the result is analytically continued to $\kappa=1$ (which corresponds to real $J$) something novel occurs: the complexity has a finite value at $a=1$. Since the $a$-dependence gives a |