The characteristic-polynomial conjecture for the Hermitian Killing form of a harmonic polynomial

Let $n$ be a positive integer and let $a,b\in\mathbb{C}$ satisfy $|a|\neq 1$. Set $h(z,w)=z^n+aw^n+b$ and $h^{\ast}$ for the polynomial obtained by conjugating the coefficients and variables, and consider the quotient

$$\mathbb{C}[z,w]/\langle h,h^{\ast}\rangle.$$

With respect to the basis $\lbrace [z^{\alpha}w^{\beta}]\mid \alpha,\beta<n\rbrace$, let $q_{\mathbb{C}}(\lambda)$ be the characteristic polynomial of the matrix representing the Hermitian Killing form. The characteristic-polynomial conjecture. Computations suggest that

$$q_{\mathbb{C}}(\lambda)=\frac{(\lambda-n^2)(((|a|^2-1)\lambda\pm n^2|b-a\overline{b}|)^{n-1})(((|a|^2-1)^2\lambda-n^2|b-a\overline{b}|^2)^{\frac{n(n-1)}{2}})(((|a|^2-1)^2\lambda+n^2|b-a\overline{b}|^2)^{\frac{(n-1)(n-2)}{2}})}{(|a|^2-1)^{2n(n-1)}}.$$

Consequently, when $b\neq 0$, the difference between the numbers of positive and negative roots is $n$, while when $b=0$ the polynomial reduces to $q_{\mathbb{C}}(\lambda)=\lambda^{n^2-1}(\lambda-n^2)$ and that difference is $1$. This proposed formula is motivated by computed examples and is consistent with the preceding proposition, which counts the solutions of the harmonic equation; its general validity is not established in the supplied text.