Polynomials

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 October 2009

Polynomials

Let $𝔽$ be a field. If $a_{0}, a_{1}, a_{2}, \dots \in 𝔽$ use the notation $a_{0} + a_{1} x + a_{2} x^{2} + \dots = \sum_{i \in ℤ_{\geq 0}} a_{i} x^{i} .$

The polynomial ring is the set $𝔽 [x] = \{\sum_{i \in ℤ_{\geq 0}} a_{i} x^{i} | a_{i} \in 𝔽 and all but a finite number of the a_{i} are equal to 0\}$ with operations given by $(\sum_{i \in ℤ_{\geq 0}} a_{i} x^{i}) + (\sum_{i \in ℤ_{\geq 0}} b_{i} x^{i}) = (\sum_{i \in ℤ_{\geq 0}} (a_{i} + b_{i}) x^{i})$ and $(\sum_{i \in ℤ_{\geq 0}} a_{i} x^{i}) (\sum_{j \in ℤ_{\geq 0}} b_{j} x^{j}) = (\sum_{k \in ℤ_{\geq 0}} c_{k} x^{k}), where c_{k} = \sum_{i + j = k} a_{i} b_{j} .$

Let $a \in 𝔽$ . The evaluation homomorphism is $\begin{array}{rcl} {ev}_{a} : 𝔽 [x] & ⟶ & 𝔽 \\ p (x) & ⟼ & p (a) \end{array}$ where $p (a) = p_{0} + p_{1} a + p_{2} a^{2} + \dots if p (x) = p_{0} + p_{1} x + p_{2} x^{2} + \dots .$

Let $p (x) \in 𝔽 [x]$ , $p (x) = p_{0} + p_{1} x + p_{2} x^{2} + \dots$ . The degree $\deg (p (x))$ of $p (x)$ is the maximal nonnegative intger $d$ such that $p_{d} \neq 0$ .

The field of rational functions in $x$ is the set $𝔽 (x) = \{\frac{a (x)}{b (x)} | a (x), b (x) \in 𝔽 [x], b (x) \neq 0\},$ with $\frac{a (x)}{b (x)} = \frac{c (x)}{d (x)}, if a (x) d (x) = b (x) c (x),$ and with operations given by $\frac{a (x)}{b (x)} + \frac{c (x)}{d (x)} = \frac{a (x) d (x) + b (x) c (x)}{b (x) d (x)} and \frac{a (x)}{b (x)} \times \frac{c (x)}{d (x)} = \frac{a (x) c (x)}{b (x) d (x)} .$

The ring of formal power series in $x$ is [???] CHANGED [ ] TO [[ ]] $𝔽 [[x]] = \{\sum_{i \in ℤ_{\geq 0}} a_{i} x^{i} | a_{i} \in 𝔽\},$ with operations given by $(\sum_{i \in ℤ_{\geq 0}} a_{i} x^{i}) + (\sum_{i \in ℤ_{\geq 0}} b_{i} x^{i}) = (\sum_{i \in ℤ_{\geq 0}} (a_{i} + b_{i}) x^{i})$ and $(\sum_{i \in ℤ_{\geq 0}} a_{i} x^{i}) (\sum_{j \in ℤ_{\geq 0}} b_{j} x^{j}) = (\sum_{k \in ℤ_{\geq 0}} c_{k} x^{k}), where c_{k} = \sum_{i + j = k} a_{i} b_{j} .$

Examples.

	$\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots .$
	$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots = \sum_{i \in ℤ_{\geq 0}} \frac{x^{i}}{i!} .$
	$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \dots = \sum_{i \in ℤ_{\geq 0}} {(-1)}^{i} \frac{x^{(2 i + 1)}}{(2 i + 1)!} .$
	$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \dots = \sum_{i \in ℤ_{\geq 0}} {(-1)}^{i} \frac{x^{2 i}}{(2 i)!} .$
	$\ln (1 - x) = x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + \dots = \sum_{i \in ℤ_{> 0}} \frac{x^{i}}{i} .$

The Laurent polynomials in $x$ is the set [???] CHANGED ( ) TO (( )) $𝔽 ((x)) = \{\frac{a (x)}{b (x)} | a (x), b (x) \in 𝔽 [[x]], b (x) \neq 0\},$ with $\frac{a (x)}{b (x)} = \frac{c (x)}{d (x)}, if a (x) d (x) = b (x) c (x),$ and with operations given by $\frac{a (x)}{b (x)} + \frac{c (x)}{d (x)} = \frac{a (x) d (x) + b (x) c (x)}{b (x) d (x)} and \frac{a (x)}{b (x)} \times \frac{c (x)}{d (x)} = \frac{a (x) c (x)}{b (x) d (x)} .$

$𝔽 [x]$ is an integral domain.
$𝔽 [[x]]$ is an integral domain.

The invertible elements of $𝔽 [x]$ are invertible elements of $𝔽$ .
The invertible elements of $𝔽 [[x]]$ are $a_{0} + a_{1} x + a_{2} x^{2} + \dots \in 𝔽 [[x]]$ with $a_{0}$ invertible in $𝔽$ .

𝔽 ((x)) = \{x^{k} p (x) | p (x) \in 𝔽 [[x]], p_{0} \neq 0\}

Let $p (x) \in 𝔽 ((x))$ . The order $ν (p (x))$ of $p (x) = \sum_{l \in ℤ} p_{l} x^{l}$ is the minimal integer $l$ such that $p_{l} \neq 0$ . The order function $ν : 𝔽 ((x)) \to ℤ$ is a normalised discrete valuation (see [BouC] Ch. VI §3 no.6 def.3). [???] INCLUDE THIS COMMENT/REFERENCE?

References

[BG] A. Braverman and D. Gaitsgory , Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)