This is a neat way! 

But why do you need the "[];", why does that prevent the 'overflow'?
And why you want the forget commands?

Care for brackets

And for (proper) coding you may wish to test like "if(one IsNumber) else ..."

'diff(E, b)';
F:=eval(%, data): # the data, or use subs 
plot(%, b=0 .. 10, smartview=false);

This shows where a (or the?) zero may be located and then help Maple:

fsolve(F, b = 2);

                           1.37685744189333

eval(%%, b=%);
                                      -13
                            -0.4635 10

Not very algorithmical - but that quite often works for a guess.

+- 6.56869087567479 * I

Eq:=eval(DispersionEq, [Zeta = 0.25e-1, nu = .3, rho = .128, gamma__o = 3.4,
                        Omega_max = 1.2, lambda = 0, Omega = .6]);

[RootFinding:-Analytic(Eq, k=-5-5*I .. 5+5*I)]:
sols:=sort(%);

  [-.460809992756477*I, .460809992756477*I,
   -3.05238710098628-.219398452118226*I, -3.05238710098628+.219398452118226*I,
    3.05238710098628-.219398452118226*I,  3.05238710098628+.219398452118226*I]

or use DirectSearch as done by Markiyan Hirnyk (I used Digits:=15)

@Kitonum 

Here is my variation on your solution. One can use the simplex package to find the hypercube.

MP_linear_program.mws

MP_linear_program.pdf

Ok - but I think that is no longer complete, [F = 1, H = 1, J = 1, K = 1, L = 15] (the first one from your other approach using nested loops) is not covered

@Kitonum But you had 150565 solutions. I think you need more combinations

@Alejandro Jakubi 

It may be a mtter of taste, but for the initially shown plot I would prefer some kind of legend (instead of writing text into a plot). Actually I would try a 3-dim plot first, showing the projection as second step.

You are right, sorry

I have some doubts: the parameters are all positive. Now if 0 < b then
all summands are squared Reals. So Sum = 0 if all summands are zero.

a*BesselJ(lambda,(Omega^2*g^2+k^2)^(1/2)) +
b*BesselJ(lambda+1,(Omega^2*g^2+k^2)^(1/2))

is the form of your equation and a,b have something in common. I have not
looked much closer (try to use simplify(%, size)), but since those Bessel
are periodic there may be many solution. Also I have not tried to make use
of x*BesselJ(v,x): diff(%, x): collect(%, BesselJ); Perhaps it is best to
look at the origin for the equation.

And, of course, I would expect 'good' answers for numerical values only
(just as Carl Love)

eq2=eval(eq2a, h=1);

                        a b c + a = a b c + a

Math: Homogenize <--> h = 1

Personally I consider it as vandalism to destroy initial questions.

And thus restore it (the web does not forget ...)

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1. Do Hilbert series function classify all or only some type or some form of ideals?
2. What can be further do after classified by hilbert series? I thought to use intersection of ideals. there is a need to use another dictionary to store all combination of intersection of ideals
3. which invariant can classify ideals into exact sequence or sequence?

Bonus 1: how to think out the steps in algorithm of hilbert series? (this is not essential to answer)

Bonus 2: Under unique factorization domain, there are atomic domain or GUFD and further to t-UCFD or pre-Schreier domain in paper non-atomic unique factorization in integral domain, do these domains have corresponding invariants, what are their algorithm?

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Anyway (and that will be all) you may search for the buzz words "Hilbert Scheme" and "Deformation Theory".