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 May 1st, 2019, 10:17 PM #1 Newbie   Joined: May 2013 Posts: 3 Thanks: 0 Definition of Formal power series in m indeterminates over R I could not understrand the following definition for formal power series over $m$ indeterminates, over the commutative ring $R$: *I do understand:* We set $R[\![X_{1},...,X_{m}]\!]:=(R^{(\mathbb{N}^{m})},+,.)$, where $+$ and $.$ are as in: $(p+q)_{\alpha}:=p_{\alpha}+q_{\alpha}$, $\alpha\in\mathbb{N}$ and $(pq)_{\alpha}:=\sum_{\alpha<\beta}$ $p_{\beta}q_{\alpha - \beta}$, $\alpha\in\mathbb{N}$. *But I get stuck in the following definition:* Set $X:=(X_{1},...,X_{m})$ and , for $\alpha\in\mathbb{N}$, denote by $X^{\alpha}$ the formal power series (that is, the function $\mathbb{N}^m\rightarrow R$) such that: $X_{\beta}^{\alpha}:=\lbrace 1$ for $\beta = \alpha,0$ for $\beta\ne\alpha\rbrace$, $\beta\in\mathbb{N}$ Then each $p\in R[\![X_{1},...,X_{m}]\!]$ can be written uniquely in the form $p=\sum_{\alpha\in\mathbb{N}^m}p_{\alpha}X^{\alpha }$ I could not build an example to visualize this, unlike the case for $m=1$ that was clear: by making $X=(1,0,...,0,..), X^2=(0,1,...,0,..)$ and so on. Tags definition, formal, indeterminates, power, series Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post 667 Number Theory 41 December 7th, 2013 09:27 AM asifrahman1988 Calculus 1 March 17th, 2013 03:22 AM themathlearner Abstract Algebra 1 November 28th, 2012 07:39 PM g0bearmon Real Analysis 2 May 22nd, 2012 01:10 PM g0bearmon Calculus 1 December 31st, 1969 04:00 PM

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