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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: enter image description here 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

  1. 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
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    Sure! I shall add the context and reference. – 2017-01-31
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    @Winther I have added the details as you suggested. Thanks. – 2017-01-31
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    The paper has a quite long and detailed derivation (and it's years since I did anything resembling stuff like this) so I don't know if this will help, but it looks like you can use the same method as in this paper. Note that there is no $x_i^2$ term in their integral (6) either until they average over $J_{ij}$, i.e. computing gaussian integrals on the form $\int e^{-AJ_{ij}^2 + BJ_{ij}}{\rm d}J_{ij} \sim e^{B^2}$ (note the integral is wrt $J_{ij}$), and since $B$ will contain linear terms in $x_i$ you will end up with $e^{-Cx_i^2}$ terms which leads to gaussian integrals for the $x_i$'s. – 2017-01-31
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    Exactly. But the problem is that I do not want to average over the $J_{ij}$. Is there any workaround? – 2017-01-31
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    I see. I don't know enough about this to give any sensible pointers on that. Hopefully someone with a more detailed knowledge of these kinds of systems / calculations will see this question and be able to point you in the right direction. – 2017-01-31
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    No problem, thanks! – 2017-01-31

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