Involutions of a Polynomial Ring and the Symmetric Part

Introduction

The polynomial ring $R[x]$ over a commutative ring $R$ with an involution $\sigma$ carries involutions obtained by sending $x$ to a polynomial $f$ and the coefficients by $\sigma$; the smallest interesting one is $x \mapsto -x$, whose fixed subring is $R[x^2]$ and which turns $R[x]$ into a $\mathbb{Z}/2$-graded ring with even part $R[x^2]$ and odd part $xR[x^2]$. The general involution of $R[x]$ that restricts to $\sigma$ on the coefficients is determined by $f$, it is an involution exactly when $f \circ f = x$ and $f$ is compatible with $\sigma$, and the affine ones $f(x) = ax+b$ are classified by the two conditions $a\sigma(a) = 1$ and $\sigma(a)b + \sigma(b) = 0$; the map $x \mapsto -x$ is the case $a = -1$ and $b$ a fixed element, and the identity is the case $a = 1$.

This article treats those involutions, the symmetric part they define, and the graded decomposition that the involution $x \mapsto -x$ induces. It opens with the structure of an involution of $R[x]$, continues with the symmetric part and the grading, and closes with the examples of $x \mapsto -x$, of the conjugation of $\mathbb{R}[x]$ inside $\mathbb{C}[x]$, and of the characteristic-$p$ phenomena. It assumes Polynomial Rings and Rational Functions for the polynomial ring and its degree, Involutive Rings for the involution and its fixed ring, and Ring and Field Automorphisms for the automorphisms of $R[x]$; the symmetric and skew elements are The Skew Field of a Ring with Involution, the article of this group that follows. Throughout, $R$ is a commutative ring with $1 \neq 0$ and an involution $\sigma$, and $R[x]^+$ denotes the fixed subring of an involution.

Involutions of R[x]

Definition. Let $f \in R[x]$. The involution of $R[x]$ determined by $f$ is the additive map

$$ \Bigl(\sum_i a_i x^i\Bigr)^{\sigma,f} = \sum_i \sigma(a_i)\,f(x)^i ; $$

it restricts to $\sigma$ on the constants and sends $x$ to $f$.

Theorem. The map $(\,)^{\sigma,f}$ is a ring homomorphism for every $f$, because $R[x]$ is commutative, and it is an involution if and only if

$$ \sigma^2 = \mathrm{id}, \qquad f(f(x)) = x, \qquad \text{and the coefficients of } f \text{ are twisted compatibly with } \sigma . $$

The involution is the identity on $R[x]$ exactly when $\sigma = \mathrm{id}$ and $f = x$.

Proof. The map is additive by construction, it sends $1$ to $1$, and it is multiplicative because it is an evaluation of the commutative ring $R[x]$ with the coefficients twisted: $(gh)^{\sigma,f} = g^{\sigma,f}h^{\sigma,f}$ for all $g,h$ once the coefficients of $f$ satisfy $\sigma(c)f(x) = f(x)\sigma(c)$, which holds automatically in the commutative ring. Applying the map twice gives $x \mapsto f(f(x))$ and $a_i \mapsto \sigma^2(a_i)$, so the square is the identity exactly under the stated conditions; the triviality statement is the comparison of $x$ and of the coefficients.

Proposition (affine involutions). An involution of $R[x]$ with $f(x) = ax+b$ of degree one restricts to $\sigma$ on $R$ and satisfies the conditions

$$ a\,\sigma(a) = 1, \qquad \sigma(a)\,b + \sigma(b) = 0 . $$

Conversely every pair $(a,b)$ with these two conditions and $\sigma^2 = \mathrm{id}$ determines an affine involution, $(\sum a_i x^i)^{\sigma,f} = \sum \sigma(a_i)(ax+b)^i$.

Proof. Let $\Phi$ be the twisted evaluation. Then $\Phi^2(x) = \Phi(ax+b) = \sigma(a)\Phi(x)+\sigma(b) = \sigma(a)a\,x + \sigma(a)b + \sigma(b)$; the square is the identity on $x$ exactly when $\sigma(a)a = 1$ and $\sigma(a)b+\sigma(b) = 0$, and on the coefficients it is $\sigma^2 = \mathrm{id}$. The converse is the same computation.

Corollary (the two basic affine cases). The affine involution $x \mapsto -x+b$ is the case $a = -1$, for which $\sigma(a)b+\sigma(b) = -b+\sigma(b)$ vanishes exactly when $b$ is fixed; the identity is the case $a = 1$ and $b$ with $2b = 0$. When $2$ is invertible in $R$, the affine involutions are exactly $x \mapsto -x+b$ with $b \in R^\sigma$, together with the identity.

Proof. For $a = -1$ the first condition is $(-1)(-1) = 1$, automatic, and the second is $-b+\sigma(b) = 0$; for $a = 1$ the second is $2b = 0$. When $2$ is invertible the only solution of $2b = 0$ is $b = 0$, giving the identity.

Theorem (degree two and beyond). If $R$ is an integral domain and $\deg$ is multiplicative on $R[x]$, then every involution of $R[x]$ restricting to $\sigma$ on the coefficients is affine: $\deg(f\circ f) = (\deg f)^2 = 1$ forces $\deg f = 1$. Over a ring with zero divisors and nilpotents there are non-affine involutions, the smallest being $x \mapsto x + t$ in $R[x]$ with $t^2 = 0$ and $t$ fixed, which has order two.

Proof. $R[x]$ is an integral domain when $R$ is, and the degree of a composition of nonconstant polynomials is the product of the degrees; $\deg(\mathrm{id}) = 1$ gives $(\deg f)^2 = 1$, so $f$ is linear. For the example, $(x+t)\circ(x+t) = x+2t+t^2 = x$ when $2t = 0$ and $t^2 = 0$, and the twisted evaluation by $\sigma = \mathrm{id}$ has order two.

The Symmetric Part and the Grading

Theorem. The involution $x \mapsto -x$ of $R[x]$ has fixed subring

$$ R[x]^{+} = R[x^2] = \{g(x^2) : g \in R[x]\}, $$

the polynomials in $x^2$, and it induces the $\mathbb{Z}/2$-grading

$$ R[x] = R[x^2] \oplus x\,R[x^2], $$

with even part $R[x^2]$ and odd part $xR[x^2]$. The symmetric part is the even part and the skew part is the odd part.

Proof. A polynomial $\sum a_i x^i$ is fixed by $x \mapsto -x$ exactly when the coefficients of the odd powers vanish, that is, when it is a polynomial in $x^2$; the direct sum is the grouping of the even and the odd powers, and the product of an element of $x^iR[x^2]$ and one of $x^jR[x^2]$ lies in $x^{i+j}R[x^2]$, which is the grading. The identification of the eigenspaces is the additive decomposition of Involutive Rings.

Corollary (the affine case as a translation). For the involution $x \mapsto -x+b$ with $b \in R^\sigma$ and $2$ invertible, the translation $y = x - b/2$ carries it to $y \mapsto -y$, and the fixed subring is $R[(x-b/2)^2]$, with the grading $R[x] = R[(x-b/2)^2] \oplus (x-b/2)R[(x-b/2)^2]$.

Proof. $x \mapsto -x+b$ is the reflection of the line about $b/2$, so the substitution $y = x-b/2$ makes it $y \mapsto -y$; the fixed subring and the grading are those of the previous theorem transported by the automorphism $x \mapsto x+b/2$ of $R[x]$, which is legitimate because $b$ is fixed.

Remark. For the involution $x \mapsto -x$ with $\sigma = \mathrm{id}$ the fixed subring $R[x^2]$ is a polynomial ring in $x^2$ and $R[x]$ is free over it with basis $1, x$; for a general affine involution the fixed subring is the intersection of the coefficient conditions with the vanishing of the odd powers of the translate and need not be a polynomial ring in one element.

Examples

(a) The basic case. $R[x]$ with $\sigma = \mathrm{id}$ and $x \mapsto -x$ has fixed subring $R[x^2]$ and the grading $R[x] = R[x^2]\oplus xR[x^2]$; this is the model of the $\mathbb{Z}/2$-grading by parity of degree.

(b) The conjugation of the real polynomials. $R = \mathbb{C}$, $\sigma$ the conjugation, $f = x$: the involution is the coefficientwise conjugation, its fixed subring is $\mathbb{R}[x]$ inside $\mathbb{C}[x]$, and the pair is the quadratic extension $\mathbb{C}[x]/\mathbb{R}[x]$.

(c) The conjugation twisted. $R = \mathbb{C}$, $\sigma$ the conjugation, $f = -x$: the fixed subring consists of the polynomials with real coefficients in the even powers and purely imaginary coefficients in the odd powers, that is the ring $\mathbb{R}[x^2] \oplus i\,x\,\mathbb{R}[x^2]$, a free $\mathbb{R}[x^2]$-module of rank two.

(d) The involution of the circle. $R = \mathbb{R}$, $f = -x$: the fixed subring $\mathbb{R}[x^2]$; the ring $\mathbb{R}[x]/(x^2+1) = \mathbb{C}$ carries the induced conjugation, which is the involution $i \mapsto -i$, and the fixed field is $\mathbb{R}$.

(e) Characteristic two. $R = \mathbb{F}_2$ with $\sigma = \mathrm{id}$: the involution $x \mapsto -x$ is the identity, since $-x = x$; the affine involutions satisfy $2b = 0$ automatically, and $x \mapsto x+1$ is an involution whose fixed subring is the constants $\mathbb{F}_2$, because a nonconstant polynomial cannot satisfy $g(x+1) = g(x)$ over $\mathbb{F}_2$.

(f) The power series case. The same computation gives the involution $t \mapsto -t$ of $k[[t]]$ with fixed subring $k[[t^2]]$ and the grading $k[[t]] = k[[t^2]]\oplus t\,k[[t^2]]$; this is the local analogue of (a) and is used in Involutive Local Rings.

Summary

An involution of $R[x]$ that restricts to $\sigma$ on the coefficients is determined by a polynomial $f$ and sends $\sum a_i x^i$ to $\sum \sigma(a_i)f(x)^i$; it is an involution exactly when $\sigma^2 = \mathrm{id}$ and $f \circ f = x$ with the coefficient compatibility, and it is an automorphism because $R[x]$ is commutative. The affine involutions $f(x) = ax+b$ are classified by $a\sigma(a) = 1$ and $\sigma(a)b+\sigma(b) = 0$; over a field $R$ of characteristic zero, or over any integral domain with a multiplicative degree, every involution is affine, and when $2$ is invertible the only ones are the identity and $x \mapsto -x+b$ with $b$ fixed. Over a ring with nilpotents there are non-affine involutions, such as $x \mapsto x+t$ with $t^2 = 0$.

The involution $x \mapsto -x$ is the model of the $\mathbb{Z}/2$-grading: its fixed subring is $R[x^2]$, the symmetric part is the even part and the skew part is the odd part, and $R[x] = R[x^2]\oplus xR[x^2]$. The affine involution $x \mapsto -x+b$ is the translate of this one by $b/2$, with fixed subring $R[(x-b/2)^2]$, and the conjugation of $\mathbb{C}[x]$ over $\mathbb{R}[x]$, the twisted case $x \mapsto -x$, the circle example $\mathbb{R}[x]/(x^2+1)$ and the power series analogue are the standard instances.

Summary of Notation

Symbol	Meaning
$R$, $\sigma$	Commutative ring and its involution
$f \in R[x]$	The polynomial to which $x$ is sent
$(\sum a_i x^i)^{\sigma,f} = \sum \sigma(a_i)f^i$	The involution determined by $f$
$f\circ f = x$, $\sigma^2 = \mathrm{id}$	Conditions for the map to be an involution
$f = ax+b$, $a\sigma(a) = 1$, $\sigma(a)b+\sigma(b) = 0$	Classification of the affine involutions
$x \mapsto -x$	The basic involution; fixed subring $R[x^2]$
$R[x] = R[x^2]\oplus xR[x^2]$	The induced $\mathbb{Z}/2$-grading
$x \mapsto -x+b$, $b \in R^\sigma$	Affine case, translate of $x \mapsto -x$ by $b/2$
$\mathbb{R}[x] \subset \mathbb{C}[x]$	Conjugation example
$\mathbb{F}_2[x]$, $x \mapsto x+1$	Characteristic-two involution; dual numbers