Timeline for Galois invariants in a ring of fractional power series over a finite field

6 events

when toggle format	what		by	license	comment
Apr 25, 2012 at 16:33	comment	added	Jared Weinstein		Dror: This is top-rate work (and certainly answer-worthy). I suspected there wouldn't be a clean formula for $\pi$, but at least now I know there's an algorithm.
Apr 25, 2012 at 15:33	comment	added	Dror Speiser		For example, for $q=3$ we have $\pi=-x^{2/3}-x^{10/9}+x^{14/9}+x^{46/27}-x^2-x^{58/27}+...$
Apr 25, 2012 at 15:17	comment	added	Dror Speiser		Let $R_1=\mathbb{F}_q[[X]],R_2=R_1[[Y]]$. Set $h_1=-X^{q-1}\in R_1$, so that $\pi=h_1(x_1)$. Let $g_2(Y)=h_1(X^q+YX)\in R_2$. Then $\lim_{n\rightarrow\infty}g_2^{(n)}(0)$ converges in $R_1$ (where $g_n^{(n)}$ is $n$-fold composition), and denote this limit by $h_2$. We have $h_2(x_2)=\pi$. We can continue similarly: $g_{i+1}(Y)=h_i(X^q+YX)$, $h_{i+1}=\lim_{n\rightarrow\infty}g_{i+1}^{(n)}(0)$, $h_{i+1}(x_{i+1})=\pi$. Magma computations show that $H=\lim_{n\rightarrow\infty} h_n(X^{1/q^{n}})$ converges in $\mathbb{F}_q[[X^{1/q^\infty}]]$. So finally $H(x)=\pi$.
Apr 25, 2012 at 7:18	comment	added	S. Carnahan♦		@Will: The power series $\sum a_n \pi^n$ preserves constants, and sends $x$ to $\sum a_n x^{q^n}$.
Apr 25, 2012 at 5:54	comment	added	Will Sawin		I don't understand the group action you are describing, could you elaborate on exactly what it does?
Apr 25, 2012 at 1:55	history	asked	Jared Weinstein	CC BY-SA 3.0