PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.ChargeBalance

Let $R$ be a commutative ring and $\sigma$ be a set of indices for variables. For any multivariate polynomial $f \in R[X_i]_{i \in \sigma}$, any scaling factors $d_i \in R$ (for $i \in \sigma$), and any monomial $X^m = \prod_{i \in \sigma} X_i^{m_i}$ (where $m: \sigma \to \mathbb{N}$ is a finitely supported exponent vector), the coefficient of $X^m$ in the polynomial obtained by substituting $d_i X_i$ for each $X_i$ is given by: $$\text{coeff}(X^m, f(d_i X_i)) = \left( \prod_{i \in \sigma} d_i^{m_i} \right) \cdot \text{coeff}(X^m, f)$$

Let $K$ be a field and $c \in K$ be a non-zero element of infinite order (i.e., $c^n = 1$ if and only if $n = 0$ for $n \in \mathbb{Z}$). Let $f \in K[X_i]_{i \in \sigma}$ be a multivariate polynomial where each variable $X_i$ is assigned an integer weight (or charge) $w_i \in \mathbb{Z}$. If $f$ is invariant under the transformation $X_i \mapsto c^{w_i} X_i$, then for any monomial $X^m = \prod_{i} X_i^{m_i}$ with a non-zero net charge $\sum_i m_i w_i \neq 0$, the coefficient of $X^m$ in $f$ must be zero.