# name for a degree-like invariant of a power series over a commutative ring

Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$f = \sum_{i=1}^n a_i u_i X^i,$$ where $a_i \in R$ and $u_i$ is a unit in $R[\![X]\!]$. I call the smallest such $n$ for which such an expression is possible the pseudodegree of $f$.

At least, that's what I and a coauthor called it in a recent preprint. It seemed reasonable, since if one makes the same definition in the polynomial ring $R[X]$, this concept coincides with the usual notion of the degree of a polynomial, at least when $R$ is reduced. Also, like the degree of a polynomial, the pseudodegree of $f$ is at least as big as one less than the minimal number of generators of the ideal of $R$ that is generated by the coefficients of $f$.

But this must have a name somewhere, right? Can anyone point to an article, book, preprint or the like where this concept has been used? What else can be said about pseudodegree?

• A trivial comment. Pseudodegree may not coincide with the usual degree in polynomial rings if the base ring had nilpotents. – Mohan Feb 15 '14 at 17:03
• Oh, good point; I'll make the corresponding edit. – Neil Epstein Feb 15 '14 at 19:31
• I see. I figured there was something I was missing. – Ben Webster Feb 16 '14 at 19:20
• If $f=\sum_{i=0}^\infty a_i X^i$, and $I_n=(a_0,\dots,a_n)$, then this is the $n$ at which the increasing sequence of ideals $I_n$ stabilizes, right? – Will Sawin Feb 18 '14 at 3:08
• Question: is the psuedo degree multiplicative? is there a counter-example? – user82090 Oct 30 '15 at 9:57