# Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$
Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x_1, \dotsc, x_n] \to A$. For $f \in A$, define $\mathrm{deg}(f)$ to be the minimum of $\mathrm{deg}(g)$, where $g$ ranges over all polynomials in $k[x_1, \dotsc, x_n]$ such that $\phi(g) = f$. [Note: by $\mathrm{deg}(g)$, I mean the degree of the highest-degree monomial, where $\mathrm{deg}(x_1^{i_1} \dotsm x_n^{i_n}) = i_1 + \dotsb + i_n$.] If it is helpful, we can assume $A$ is an integral domain, even integrally closed if necessary.
Let $u \in A^*$ be a unit such that $\mathrm{deg}(u) > 0$, or equivalently, $u \not\in k^*$. Is it necessarily true that $\deg(u^n) \to \infty$ as $n \to \infty$?
Thoughts: If we have a monomial order that respects degree (such as grlex or grevlex, but not lex), and take a Groebner basis of $\ker(\phi)$, then we see that powers of $u$ remain predictable as long as their leading terms fall outside the ideal generated by the leading terms of the groebner basis (aka, the initial ideal).