Rational formal power series, proper presentations and reduced denominators

Let $R$ be a commutative ring. A formal power series $F\in R\mathrm{⟦}x\mathrm{⟧}$ (Formal power series over a commutative ring and the coefficient-extraction functional $[x^n]$) is rational if there are polynomials $P,Q\in R[x]$ such that $Q(0)$ is a unit and

$QF=P$

in $R\mathrm{⟦}x\mathrm{⟧}$. Equivalently, $F=P\mathrm{/}Q$, where $Q^{-1}$ is the unique formal inverse supplied by A formal power series is a unit exactly when its constant coefficient is a unit.

Now let $R=K$ be a field. Multiplying numerator and denominator by $Q(0{)}^{-1}$ gives a normalised presentation with $Q(0)=1$. Such a presentation is proper when $P=0$ or $degP<\mathrm{deg}Q$ (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree). It is reduced when $P$ and $Q$ have no common nonunit factor. A polynomial series has the reduced presentation $P\mathrm{/}1$, and the zero series has reduced presentation $0\mathrm{/}1$.

A polynomial $Q$ occurring in a normalised reduced presentation is called a reduced denominator of $F$. The minimal-order theorem will show that all reduced denominators of one series have the same degree.