Polynomiality and monodromy invariance of the Novikov-algebra coordinates

Let $\boldsymbol{v}=(v^1,\ldots,v^n)$ be the coordinates associated with a Novikov algebra satisfying the conditions in Assumption 2, and let $\boldsymbol{u}(\boldsymbol{v})=(u^1(\boldsymbol{v}),\ldots,u^n(\boldsymbol{v}))$ be the corresponding functions. Suppose that $\lambda_i\in\mathbb{N}_{>0}$ for all $i=1,\ldots,n$. Writing $\mathcal{W}(\mathcal{A})$ for the monodromy group, the functions $u^i(\boldsymbol{v})$ are polynomial and invariant under this group:

$$u^i(\boldsymbol{v})\in\mathbb{C}^{\mathcal{W}(\mathcal{A})}[v^1,\ldots,v^n].$$

Polynomiality and monodromy-invariance conjecture. Under these hypotheses, every function $u^i(\boldsymbol{v})$ belongs to the invariant polynomial ring $\mathbb{C}^{\mathcal{W}(\mathcal{A})}[v^1,\ldots,v^n]$. The claim concerns the polynomiality of the coordinate functions and their invariance under the monodromy action.

The conjecture is motivated by examples in which subtle cancellations remove apparent rational dependence on the variable $v^n$. Its validity depends on the stated conditions on the Novikov algebra, and the absence of a full classification of Novikov algebras leaves the general claim open.