Studying a certain system of hypergeometric partial differential equations, I came across the proof that the $ \mathcal{D}$-module associated to it is holonomic. My issue it that I can't follow the proof in its entirety. First let me state the setup:
We are given $ N $ lattice points $ a_{1}, \dots a_{N} \in \mathbb{Z}^{n} $ such that their $ \mathbb{R} $-span equals $ \mathbb{R}^{n}$. Consider the ring generated by the monomials with exponents $ a_{i} $, $ R=\mathbb{C}[x^{a_{1}}, \dots , x^{a_{N}}] $ and the polynomial $ f=\sum_{i=1}^{N}\alpha_{i}x^{2a_{i}} $, where $ \alpha_{i} \in \mathbb{C}\setminus \{0\} \, \forall i \in \overline{1,n} $
The proof then boils down to showing that the only maximal ideal of $ R $ containing all the polynomials $ x_{i}\frac{\partial f }{\partial x_{i}} $ where i $\in \overline{1,n}$ is the ideal $ (x^{a_{1}}, \dots x^{a_{N}}) $.
I would appreciate if someone could point out a proof of it or some hints. Thank you!