Efficient Quotients of Non-Commutative Polynomials

We adopt the following notation:
$\operatorname{prec}_{B}u$ number of base-$B$ digits of an integer $u$, $\lfloor\log_{B}|u|\rfloor+1$ $\operatorname{prec}_{x}p$ number of coefficients of a polynomial $p$, $\operatorname{degree}_{x}p+1$ $u\mathbin{\mathrm{quo}}v,\;u\mathbin{\mathrm{rem}}v$ quotient and remainder (see below) $u\,\text{{\bf\footnotesize x}quo}\,v,\;u\,\text{{\bf\footnotesize x}rem}\,v$ left and right (pseudo)quotient and remainder, $\text{{\bf\footnotesize x}}\in\{\mathrm{l,lp,r,pr}\}$ $\operatorname{shift}_{n}v$, $\operatorname{shinv}_{n}v$ whole shift and whole shifted inverse (see below) $R[x;\sigma,\delta]$, $R[x,\delta]$ skew polynomials (see Section 6) $\,{}_{i}u$, $u_{i}$ coefficient of skew polynomial $u$ with variable powers on the left, right. $\text{{\bf\footnotesize x}shift}_{n}v$, $\text{{\bf\footnotesize x}shinv}_{n}v$ left and right whole shift and shifted inverse, $\text{{\bf\footnotesize x}}\in\{\mathrm{l,r}\}$ (see Section 6) $X_{(i)}$ value of $X$ at $i^{th}$ iteration
The “$\operatorname{prec}$” notation, standing for “precision”, means the number of base-$B$ digits or polynomial coefficients. It is similar to that of [4], where it is used to present certain algorithms generically for integers and polynomials. In particular, if we take integers to be represented in base-$B$, i.e. for any integer $u\neq 0$ there is $h=\operatorname{prec}_{B}(u)-1$, such that

then integers base-$B$ behave similarly to univariate polynomials with coefficients $u_{i}$, but with carries complicating matters.

The notion of integer quotients and remainders can be extended to more general rings. For a Euclidean domain $D$ with valuation $N:D\rightarrow\mathbb{Z}_{\geq 0}$, such that for any $u,v\in D,v\neq 0$, there exist $q,r\in D$ such that

The value $q$ is a quotient of $u$ and $v$ and $r$ is a remainder of dividing $u$ by $v$ and we write

when these are unique. When both the quotient and remainder are required, we write $u~{}\mathrm{div}~{}v=(u~{}\mathbin{\mathrm{quo}}~{}v,u~{}\mathbin{\mathrm{rem}}~{}v)$. When $D$ is a non-commutative ring with a valuation $N$, there may exist left and right quotients such that

When these exist and are unique, we write

For certain non-commutative rings with a distance measure $\|\cdot\|$, a sequence of approximations to the inverse of $A$ may be computed via the Newton-Schulz iteration [7]

where $1$ denotes the multiplicative identity of the ring. There are several ways to arrange this expression, but the form above emphasizes that as $X_{(i)}$ approaches $A^{-1}$, the product $X_{(i)}(1-AX_{(i)})$ approaches $0$. For $\mathbb{C}^{n\times n}$ matrices, a suitable initial value is $X_{(0)}=A^{\dagger}/(n\,\mathrm{Tr}(AA^{\dagger}))$, where $A^{\dagger}$ is the Hermitian transpose.

In previous work [10] we studied the problem of efficient domain-preserving computation of quotients and remainders for integers and polynomials, then generalized these results to a generic setting. To this end, we defined the notions of the whole shift and whole shifted inverse with attention to commutative domains. We recapitulate these definitions and two results relevant to the present article.

For classical and Karatsuba multiplication it is more efficient to compute just the top part of the product in (6), omitting the lower $h$ terms, instead of shifting:

with $\text{\sc MultQuo}(a,b,n)=ab\mathbin{\mathrm{quo}}x^{n}$ computing only $\operatorname{degree}a+\operatorname{degree}b-n+1$ terms. For multiplication methods where computing only the top part of the product gives no saving, some improvement is obtained using

A suitable starting value for $w_{(0)}$ is given by Shinv0 in Section 4.

Let $u$ and $v$ be two polynomials in $R[x]$ with Euclidean norm being the polynomial degree. The left and right quotients and remainders are defined as in (2). Left and right quotients will exist provided that $v_{k}$ is invertible in $R$ and they may be computed by Algorithm 1. In the presentation of the algorithm, $\pi$ denotes a permutation on two elements so is either the identity or a transposition. The notation $\times_{\pi}$ is a shorthand for $\times\circ\pi$ so $a\times_{\pi}b=a\times b$ when $\pi$ is the identity and $a\times_{\pi}b=b\times a$ when $\pi$ is a transposition.

There are some circumstances where quotients or related quantities may be computed even if $v_{k}$ is not invertible. When $R$ is an integral domain, quotients may be computed as usual in $K[x]$ with $K$ being the quotient field of $R$. Alternatively, when $R$ is non-commutative but $v_{k}$ commutes with $v$, it is possible to compute pseudoquotients and pseudoremainders satisfying

as shown in Algorithm 2. In this case, we write

Requiring $v_{k}$ to commute with $v$ is quite restrictive, however, so we focus our attention to situations where the inverse of $v_{k}$ exists.

We now examine the notions of the whole shift and whole shifted inverse for $R[x]$ with non-commutative $R$. First consider the whole shift. Since $x$ commutes with all values in $R[x]$, we may without ambiguity take, for $u=\sum_{i=0}^{h}u_{i}x^{i}$ and $n\in\mathbb{Z}$,

That is, the fact that $R[x]$ is non-commutative does not lead to left and right variants of the whole shift.

We now come to the main point of this section and show $\operatorname{shinv}$ is well-defined when $R$ is non-commutative.

• Proof.Let $q_{\text{\sc l}}=x^{h}\operatorname{lquo}v$ and $q_{\text{\sc r}}=x^{h}\operatorname{rquo}v$. If $h<k$, then $q_{\text{\sc l}}=q_{\text{\sc r}}=0$. Otherwise, both $q_{\text{\sc l}}$ and $q_{\text{\sc r}}$ have degree $h-k\geq 0$ so

  	$\displaystyle v_{k}\,{q_{\text{\sc l}}}_{h-k}$	$\displaystyle=1$	$\displaystyle{q_{\text{\sc r}}}_{h-k}\,v_{k}$	$\displaystyle=1$		(8)
  	$\displaystyle\sum_{j=M}^{k}v_{j}\,{q_{\text{\sc l}}}_{i+k-j}$	$\displaystyle=0$	$\displaystyle\sum_{j=M}^{k}{q_{\text{\sc r}}}_{i+k-j}\,v_{j}$	$\displaystyle=0,\quad 0\leq i<h-k,$		(9)

  where $M=\max(0,i-h+2k)$. We show by induction on $i$ that ${q_{\text{\sc l}}}_{i}={q_{\text{\sc r}}}_{i}$ for $0\leq i\leq h-k$. Since $v_{k}$ is invertible, (8) and (9) give

  $${q_{\text{\sc l}}}_{h-k}={q_{\text{\sc r}}}_{h-k}=v_{k}^{-1}$$

  (10)

  and

  $\displaystyle{q_{\text{\sc l}}}_{i}$

  $\displaystyle=-\sum_{j=M}^{k-1}v_{k}^{-1}\,v_{j}\,{q_{\text{\sc l}}}_{i+k-j}$

  $\displaystyle{q_{\text{\sc r}}}_{i}$

  $\displaystyle=-\sum_{j=M}^{k-1}{q_{\text{\sc r}}}_{i+k-j}\,v_{j}\,v_{k}^{-1},\quad 0\leq i<h-k.$

  (11)

  Equation (10) gives the base of the induction. Now suppose ${q_{\text{\sc l}}}_{i}={q_{\text{\sc r}}}_{i}$ for $N<i\leq h-k$. Then for $i=N\geq 0$ equation (11) gives

  	$\displaystyle{q_{\text{\sc l}}}_{N}$	$\displaystyle=-\sum_{j=M}^{k-1}v_{k}^{-1}\,v_{j}\,{q_{\text{\sc l}}}_{N+k-j}=-\sum_{j=M}^{k-1}v_{k}^{-1}\,v_{j}\,{q_{\text{\sc r}}}_{N+k-j}$
  		$\displaystyle=-\sum_{j=M}^{k-1}v_{k}^{-1}\,v_{j}\left(-\sum_{\ell=M}^{k-1}\,{q_{\text{\sc r}}}_{N+k-j+k-\ell}v_{\ell}\,v_{k}^{-1}\right)$
  		$\displaystyle=-\sum_{\ell=M}^{k-1}\left(-\sum_{j=M}^{k-1}v_{k}^{-1}\,v_{j}\,{q_{\text{\sc r}}}_{N+k-j+k-\ell}\right)\,v_{\ell}\,v_{k}^{-1}=-\sum_{\ell=M}^{k-1}{q_{\text{\sc r}}}_{N+k-j}\,v_{\ell}\,v_{k}^{-1}={q_{\text{\sc r}}}_{N}.$

  $\square$

Thus we may write $\operatorname{shinv}_{h}v$ without ambiguity in the non-commutative case, i.e

We consider computing the left and right quotients in $R[x]$ from the whole shifted inverse. We have the following theorem.

• Proof.Consider first the right quotient. It is sufficient to show

  $$u=\operatorname{shift}_{-h}(u\times\operatorname{shinv}_{h}v)\times v+r_{\text{\sc r}}$$

  for some $r_{\text{\sc r}}$ with $\operatorname{degree}r_{\text{\sc r}}<k$. It is therefore sufficient to show

  $$\operatorname{shift}_{-k}u=\operatorname{shift}_{-k}\big{(}\operatorname{shift}_{-h}(u\times\operatorname{shinv}_{h}v)\times v\big{)}.$$

  (13)

  We have

  	$\displaystyle(u\times\operatorname{shinv}_{h}v)\times v$	$\displaystyle=u\times((x^{h}\operatorname{rquo}v)\times v)$		(14)
  		$\displaystyle=u\times(x^{h}-\rho),\quad\rho=0\text{ or }\operatorname{degree}\rho<k$
  		$\displaystyle=\operatorname{shift}_{h}u-u\times\rho.$
  	$\displaystyle\operatorname{shift}_{h}u$	$\displaystyle=(u\times\operatorname{shinv}_{h}v)\times v+u\times\rho.$		(15)

  Since $h\geq 0$, Theorem 3 applies and equation (15) gives

  $\displaystyle u$

  $\displaystyle=\operatorname{shift}_{-h}\big{(}(u\times\operatorname{shinv}_{h}v)\times v\big{)}+\operatorname{shift}_{-h}(u\times\rho)$

  with the degree of $\operatorname{shift}_{-h}(u\times\rho)$ less than $k$. Therefore

  	$\displaystyle\operatorname{shift}_{-k}u$	$\displaystyle=\operatorname{shift}_{-k-h}\big{(}(u\times\operatorname{shinv}_{h}v)\times v\big{)}$
  		$\displaystyle=\operatorname{shift}_{-k}\big{(}\operatorname{shift}_{-h}(u\times\operatorname{shinv}_{h}v)\times v)\big{)},$

  by Theorem 4, and we have shown equation (13) as required. The proof for $\operatorname{lquo}$ replaces equation (14) with

  $$v\times(\operatorname{shinv}_{h}v\times u)=(v\times(x^{h}\operatorname{lquo}v))\times u$$

  and follows the same lines, mutatis mutandis. $\square$

As in the commutative case, it may be more efficient to compute only the top part of the product instead of computing the whole thing then shifting away part. Now that we have shown that $\operatorname{shift}$ and $\operatorname{shinv}$ are well-defined for non-commutative $R[x]$, we next see that $\operatorname{shinv}$ may be computed by our generic algorithm.

Consider a ring of objects with elements from a ring $R$ extended by $x$, with $x$ not necessarily commuting with elements of $R$. By distributivity, any finite expression in this extended ring is equal to a sum of monomials, the monomials composed of products of elements of $R$ and $x$. To have a well-defined degree compatible with that of usual polynomials, it is required that

We call the elements of such a ring skew polynomials. Condition (17) implies that for all $r\in R$ there exist $\sigma(r),\delta(r)\in R$ such that

Therefore, to have well-defined notion of degree, the ring must be an Ore extension, $R[x;\delta,\sigma]$. Ore studied these non-commutative polynomials almost a century ago [6] and overviews of Ore extensions in computer algebra are given in [1, 2]. The subject is viewed from a linear algebra perspective in [3] and the complexity of skew arithmetic is studied in [9]. The ring axioms of $R[x;\sigma,\delta]$ imply that $\sigma$ be an endomorphism on $R$ and $\delta$ be a $\sigma$-derivation, i.e. for all $r,s\in R$

Different choices of $\sigma$ and $\delta$ allow skew polynomials to represent linear differential operators, linear difference operators, $q$-generalizations of these and other algebraic systems.

Condition (18) implies that it is possible to write any skew polynomial as a sum of monomials with all the powers of $x$ on the right or all on the left. We will use the notation $u_{i}$ for coefficients of skew polynomials with all powers of the variable on the right and $\,{}_{i}u$ for coefficients with all powers of the variable on the left, e.g.