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| -rw-r--r-- | bezout.tex | 11 |
1 files changed, 5 insertions, 6 deletions
@@ -92,8 +92,7 @@ not possible. The same idea may be implemented by performing diffusion in the $J$'s, and following the roots, in complete analogy with Dyson's stochastic dynamics. -This study also provides a complement to the work on the distribution of zeroes -of random polynomials \cite{Bogomolny_1992_Distribution}. + For our model the constraint we choose $z^2=N$, @@ -113,11 +112,11 @@ Critical points are given by the set of equations: \begin{equation} \frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i, i_2\cdots i_p}z_{i_2}\cdots z_{i_p} = \epsilon z_i \end{equation} -which for given $\epsilon$ are a set pf $N$ equations (plus the constraint) of degree $p-1$. +which for given $\epsilon$ are a set of $N$ equations of degree $p-1$, to which one must add the constraint condition. +In this sense this study also provides a complement to the work on the distribution of zeroes +of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ and $p \rightarrow \infty$. -$ -Since $H$ is holomorphic, a point is a critical point of its real part if and -only if it is also a critical point of its imaginary part. The number of +Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical point of $\Im H_0$. The number of critical points of $H$ is therefore the number of critical points of $\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be interpreted as a real function of $2N$ real variables. The number of critical |