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Polynomial transformation of coefficients

Posted 4 months ago
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Given a 3D polynomial of degree $m$ and coefficients $a_{ijk}^0$, $$p(x_0, y_0, x_0) = \sum_{i,j,k}^{m} a_{ijk}^0 x_0^i y_0^j z_0^k, \quad i + j + k \le m,$$

I want to make the transformation: $$x_0 = m_{11}x_1+ m_{12}y_1 + m_{13}z_1 + b_1 \\y_0 = m_{21}x_1+ m_{22}y_1 + m_{23}z_1 + b_2 \\ z_0 = m_{31}x_1+ m_{32}y_1 + m_{33}z_1 + b_3.$$

We can write this new polynomial as, $$p(x_1, y_1, z_1) = \sum_{i,j,k}^{m} a_{ijk}^1 x_1^i y_1^j z_1^k, \quad i + j + k \le m.$$

I want to use Mathematica to come up with a general expression for the coefficient terms $a_{ijk}^1$ in front of each monomial in the resultant polynomial. In Mathematica, if I explicitly specify the degree $m$, make the substitution for $x_0, y_0, z_0$ and evaluate the original sum then I can get the output for each new coefficient. However, is there a way to evaluate this in Mathematica (or another software) in the general case, without specifying a degree?

Edit: Here is how I find the coefficients for a polynomial of degree 2:

x0 = m11*x1 + m12*y1 + m13*z1 + b1
y0 = m21*x1 + m22*y1 + m23*z1 + b2
z0 = m31*x1 + m32*y1 + m33*z1 + b3
m = 2
eqn = Sum[If[i + j + k <= m, Subscript[a, i, j, k]*x0^i*y0^j*z0^k, 0], {i, 0, m}, {j, 0, m}, {k, 0, m}]
Plus @@ MonomialList[Expand[eqn], {x1, y1, z1}]
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