Timeline for GCD in polynomial vs. formal power series rings

9 events

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May 12, 2016 at 20:50	history	edited	user91576	CC BY-SA 3.0	added 11 characters in body
May 12, 2016 at 20:23	history	edited	user91576	CC BY-SA 3.0	added 129 characters in body
May 12, 2016 at 20:21	comment	added	user91576		Right, modulo multiplication by a unit. Editing the question.
May 12, 2016 at 19:45	answer	added	Friedrich Knop		timeline score: 2
May 12, 2016 at 19:42	comment	added	Mohan		As abx says, there are some issues of comparison. But, it can be made sensible as follows. Writing $f=ah, g=bh$, with $a, b$ polynomials and $\mathrm{gcd}(a,b)=1$, suffices to show that the same holds in the power series. If $a(0)\neq 0$ or $b(0)\neq 0$, they are units in the power series and so this is alright. If both are zero, then use the fact that they form a regular sequence in the polynomial ring and so remain a regular sequence in the power series and thus have no common factor in the power series.
May 12, 2016 at 19:39	comment	added	Friedrich Knop		What is meant is: Is a gcd in $\mathbb C[x_i]$ also a gcd in $\mathbb C[[x_i]]$?
May 12, 2016 at 19:26	comment	added	abx		The g.c.d. is defined up to a unit. In $\mathbb{C}[x]$ this means up to a scalar, but in $\mathbb{C}[[x]]$ up to any formal series with nonzero constant term, so you can always find a g.c.d. of hte form $x^n$. Thus the comparison does not make much sense.
May 12, 2016 at 19:09	review	First posts
May 12, 2016 at 19:12
May 12, 2016 at 19:07	history	asked	user91576	CC BY-SA 3.0