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Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module.

We consider the two following submodules of $A$:

  1. The ideal $I$ of $\mathbb{C}[u]$ generated by polynomials which vanish on $\mathbb{C}^k \subset \mathbb{C}^n$, for some fixed subspace $\mathbb{C}^k$;
  2. The submodule $J_r := \{ u^m, m \in \mathbb{Z}^k | p(m) \geq r \}$, for some linear function $p : \mathbb{R}^k \to \mathbb{R}$, where $\mathbb{Z}^k$ is the integer lattice in $\mathbb{C}^k$, and $\mathbb{C}^k = \mathbb{R}^k \otimes \mathbb{C}$.

Suppose that the quotient algebra $$\mathbb{C}[u] / (I + J_r \cap \mathbb{C}[u])$$ is a finite dimensional vector space.

I would like to understand, or at least to have a geometrical insight of why the following holds:

Let $\mathcal{J}_r$ be the image of $J_r$ in the quotient $A / IA$, $\mathcal{J}$ be a submodule of $A /IA$ such that there exist $r_- < r_+$ with $$ \mathcal{J}_{r_+} \subset \mathcal{J} \subset \mathcal{J}_{r_-}.$$

Then there exists $q \in A / IA$ such that $q \notin \mathcal{J}$ but $u_1 q , ..., u_n q \in \mathcal{J}$.

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