# how to solve equations for polynomials coefficients?

I am new to SAGE so sorry if this question is trivial. I have this code

  R2.<a0,a1,a2,a3, a4,b0,b1,b2,b3, b4,z,w,l,E,E1,v,t> = PolynomialRing(ZZ,17)
R.<x> = PowerSeriesRing(R2)
p = a1*x -a1*z*x**2+ a3*x**3+a4*x**4 +O(x**12)
k = b1*x+b2*x**2+ b3*x**3+b4*x**4 +O(x**12)
p1 = p.derivative()
p2 = p1.derivative()
y =2/x*(z-v*x)-E
x=2/x*a1*w*x**2
g=x*(p2 +y*p-x-E1*k)
v1=g.coefficients()
solve(v1[0],a3)


I am getting this error a3 is not a valid variable.

My question is how to solve equations for polynomials coefficients?

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1

solve is supposed to have (a listof) symbolic expressions (or symbolic expressions) as first argument, and (al list of) symbolic variables as second argument. in the call solve(v1[0], a3), both arguments are polynomials :

sage: v1[0].parent()
Multivariate Polynomial Ring in a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, z, w, l, E, E1, v, t over Integer Ring
sage: a3.parent()
Multivariate Polynomial Ring in a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, z, w, l, E, E1, v, t over Integer Ring


A possible workaround is :

sage: WA=SR(v1[0]).solve(SR(a3))[0]; WA
a3 == 1/3*a1*z^2 + 1/6*E*a1 + 1/6*E1*b1 + 1/3*a1*v + 1/3*a1*w


but note that

sage: WA.rhs().parent()
Symbolic Ring


I do not understand what

 x=2/x*a1*w*x**2


is supposed to mean...

( 2020-03-22 20:01:50 +0200 )edit