Let $\mathbb{F}$ be a field. And let $\mathbb{F}(\bar x)$ be the field of rational functions over the field $\mathbb{F}$ in the indeterminate $x_1,\dots,x_n$. Namely, all the functions $f:\mathbb{F}^n\rightarrow \mathbb{F}$ such that $f$ can be written as $f(\bar x)=q(\bar x)/p(\bar x)$ for two formal polynomials $q,p\in\mathbb{F}[\bar x]$.
Question: is the representation of $f(\bar x)$ as $q(\bar x)/p(\bar x)$ unique? Namely, can we have two different pairs of $q,p$ that represent the same function $f$. I am interested both in the case that the field $\mathbb{F}$ is infinite, or finite.