I have a system of non-linear equations on $N$ variables as follows.
$m_{i} = \frac{1}{1 + e^{-f_i(m_1, m_2, ..., m_N)}}$ for $1 \le i \le N$.
where $f_i$ are 4th degree polynomials in the variables. Is there any method to calculate the number of solutions to these equations?
One method that I tried to apply was to express the 'number of solutions' as a product of Dirac Delta functions and then integrate over the $m_i$ after multiplying by a suitable Jacobian. Something like the following:
But I am at a loss after this step. How can I integrate over the extra variables $x_i$.
Please also note that I am only interested in the approximate number of solutions, not exact.
EDIT:
As @Winther correctly pointed out, this is indeed in the context of minimizing the energy function of a fully connected Ising spin glass (SK model). The above equations are obtained by putting the gradient of this function to zero.
I encountered this method in the paper [1], however I am not sure whether this is a standard way of approaching such a problem. If there are any other ways, I shall be grateful to know.
In the paper, after Equation 6, the expression is averaged over Gaussian distributed random variables $J_{ij}$ because of which the exponent has a $x_i^2$ terms (Eqn 8). This (according to me) helps in integrating out the $x_i$ by completing the square in the exponent and performing a change of variables.
However I am not averaging over the $J_{ij}$ because of which there is no $x_i^2$ term. Thus I am not able to proceed further.
Any help shall be appreciated.
Thanks
- AJ Bray, MA Moore - Journal of Physics C: Solid State Physics, 1980 http://iopscience.iop.org/article/10.1088/0022-3719/13/19/002/meta