I have the following code, which produces a list of polynomials in the
infinite number of variables e_0, e_1, ...
M.<e> = InfinitePolynomialRing(QQ, implementation="sparse")
> S.<z> = LaurentSeriesRing(M)
> Qxy.<X,Y> = PolynomialRing(QQ)
> a=var('a')
> b=var('b')
> k=var('k')
> L=[]
> n=6
> x = 1/z^2 +
> sum([(-1)^(a+2)*(2*a+1)/factorial(2*a+2)*bernoulli(2*a+2)*e[2*a+2]*z^(2*a)
> for a in [1..n]])
> y = x.derivative(z)
> v = y^2 - 4*x^3 - 20*3/factorial(4)*bernoulli(4)*e[4]*x +
> 28*5/factorial(6)*bernoulli(6)*e[6]
> coeff = v.coefficients()
> for k in [0..n]:
> a = vector(coeff[k].coefficients())
> L.append((coeff[k]*a.denominator()).polynomial())
> print L
The output looks something like this:
[-e_8 + e_4^2, -e_10 + e_6*e_4, -1382*e_12 + 2205*e_8*e_4 + 500*e_6^2 -
> 1323*e_4^3, -10*e_14 + 21*e_10*e_4 + 22*e_8*e_6 - 33*e_6*e_4^2,
> 45606*e_12*e_4 + 65000*e_10*e_6 + 42042*e_8^2 - 63063*e_8*e_4^2 -
> 71500*e_6^2*e_4, 126126*e_14*e_4 + 212828*e_12*e_6 + 309582*e_10*e_8 -
> 154791*e_10*e_4^2 - 378378*e_8*e_6*e_4 - 71500*e_6^3, 858000*e_14*e_6 +
> 1337776*e_12*e_8 - 501666*e_12*e_4^2 + 760500*e_10^2 -
> 1287000*e_10*e_6*e_4 - 693693*e_8^2*e_4 - 786500*e_8*e_6^2]
These list elements should be interpreted as relations between the
variables e_i (for even i), so that, for instance, the first one should be
interpreted as e_8 = e_4^2.
Moreover, I know that these relations should allow me to inductively
express each e_k in terms of e_4 and e_6. For instance, the second relation
shows that e_10 = e_4e_6. The third relation -1382*e_12 + 2205*e_8*e_4 +
500*e_6^2 -1323*e_4^3 = 0 allows me (using a pen and paper...) to express
e_12 only in terms of e_4 and e_6, by using e_8 = e_4^2.
What I would like is to be able, for each k in the appropriate range, to
obtain a polynomial f_k(u,v) with rational coefficients, such that f_k(e_4,
e_6) = e_k. For instance, the first relation should give me f_8(u,v) = u^2.
However, I've been having a lot of trouble figuring out how to do
substitutions in these polynomial rings in order to get what I want.
I would really appreciate it if someone could point me in the right
direction. Thank you!
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