Here is my question:

Let $B = mathbb C(x_1, x_2, x_3, x_4)$ be $mathbb Z$-graded so that each $x_i$ is homogeneous and $deg(x_1, x_2, x_3, x_4) = (1, -2, 3, -4).$

$(a)$ Find $B_0.$ Hint: This ring is generated by a finite number of monomials.

$(b)$ Give the homogeneous decomposition of the polynomial:

$$f = 4x_2^6x_3^3 + x_2x_3 – x_3x_4 + 2 x_1x_3x_4 – 5x_3^3x_4^2 – x_2^4 + x_1^3x_4 + 3x_4^2 +1 $$

Here is my answer for $(a)$

I know that I am going to put a $mathbb Z$-grading on $B$ where $B = bigoplus_{i in mathbb Z}$ and $x_i in B_{deg x_i}.$So according to the degrees we have in the question we have $x_1 in B_1, x_2 in B_{-2}, x_3 in B_3, x_4 in B_{-4}.$ Also, I know that $B_i$ is the vector space with basis ${X_1^{e_1}, X_2^{e_2}, X_3^{e_3}, X_4^{e_4}}$ where $e_1 deg x_1 + e_2 deg x_2 + e_3 deg x_3 + e_4 deg x_4 = i.$ Then I got

$$B_0 =k(X_1^2 X_2 , X_1^4 X_4, X_2^3 X_3^2, X_1 X_3 X_4, X_3^4 X_4^3, X_2 X_3^2 X_4) $$

Is my solution correct?

For $(b),$ can someone tell me a reference that teaches me how to calculate the homogeneous decomposition of a polynomial or at least give me the general procedure I should follow to calculate this.

Any help will be appreciated.