# How to analytically calculate a specific pseudoinverse for multinomial root finding

Consider the following problem: $$d_1=a_{1,1}x+a_{1,2}y+a_{1,3}x^2+a_{1,4}2xy+a_{1,5}y^2$$ $$d_2=a_{2,1}x+a_{2,2}y+a_{2,3}x^2+a_{2,4}2xy+a_{2,5}y^2$$ and suppose we can represent it as follows: $$\mathbf{d}=\mathbf{A}\mathbf{z}$$ where $$\mathbf{d}=[d_1, d_2]^T$$, $$\mathbf{z}=[x, y, x^2, 2xy, y^2]^T$$, and $$\mathbf{A}=\left[\begin{matrix}a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} & a_{1,5}\\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} & a_{2,5}\end{matrix}\right].$$ Now, how can we design a pseudoinverse to find $$\mathbf{z}$$ as $$\mathbf{z}=\mathbf{A}^+\mathbf{d}$$ Note that the elements of $$\mathbf{z}$$ are not independent and I want to find the root of the multinomial near $$x=0, y=0$$. Is there any analytical approach to calculate $$\mathbf{A}^+$$ for this problem with the given format?

NOTE: the dimension of the problem may be higher but the count of the variables always equals the count of the equations.