Skew polynomial

Skew polynomials are a class of mathematical objects. They are a generalization of ordinary polynomials with a generally non- commutative multiplication . Skew polynomials are used for algebraic modeling of differential equations and difference equations .

history

Skew polynomials were first considered by the Norwegian mathematician Øystein Ore , who mainly dealt with questions of their factorization . For this reason, some authors also refer to them as Ore polynomials .

Definitions and sentences

definition

For a ring and an endomorphism of , a - derivation is defined as a mapping of in itself with the properties ${\displaystyle R}$ ${\displaystyle \sigma }$ ${\displaystyle R}$ ${\displaystyle \sigma }$ ${\displaystyle \theta }$ ${\displaystyle R}$

${\displaystyle \theta (a+b)=\theta (a)+\theta (b)\quad {\text{und}}\quad \theta (a\cdot b)=\sigma (a)\cdot \theta (b)+\theta (a)\cdot b}$

for everyone . An example for this are the infinitely often differentiable functions on the real numbers with the identity as endomorphism and the ordinary derivative . ${\displaystyle a,b\in R}$ ${\displaystyle C^{\infty }(\mathbb {R} )}$ ${\displaystyle d/dx}$

The ring of skew polynomials in the unknown is the set of formal expressions ${\displaystyle R[\partial ;\sigma ,\theta ]}$ ${\displaystyle \partial }$

${\displaystyle p=a_{n}\partial ^{n}+a_{n-1}\partial ^{n-1}+\cdots +a_{1}\partial +a_{0},}$

with coefficients in . Is , then is the degree of , which is also called order . ${\displaystyle R}$ ${\displaystyle a_{n}\neq 0}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle (\operatorname {grad} p)}$

The addition is handled like normal polynomials . The multiplication is given by the equation

${\displaystyle \partial \cdot a=\sigma (a)\cdot \partial +\theta (a)}$

set. By requiring that the associative law and the distributive law apply, one can multiply arbitrary skew polynomials with one another.

This multiplication simulates the cascading of differential operators. If in the above example we simply designate the multiplication of links with again with , then applies to any one ${\displaystyle f\in C^{\infty }(\mathbb {R} )}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle g\in C^{\infty }(\mathbb {R} )}$

${\displaystyle \left({\frac {d}{dx}}\circ f\right)(g)={\frac {d}{dx}}(f\cdot g)={\frac {df}{dx}}\cdot g+f\cdot {\frac {dg}{dx}}=\left({\frac {df}{dx}}+f\circ {\frac {d}{dx}}\right)(g),}$

where correspondingly denotes the multiplication with the derivative of . ${\displaystyle {\tfrac {df}{dx}}}$ ${\displaystyle f}$

A formal definition (and a proof of existence) for Schiefpolynome is obtained with the aid of the ring of the Gruppenendomorphismen - module ${\displaystyle R}$

${\displaystyle R^{(\mathbb {N} )}=\left\{(a_{n})_{n\in \mathbb {N} }\mid a_{n}=0{\text{ für fast alle }}n\in \mathbb {N} \right\}.}$

Now, as in the example, one embeds in the ring of group morphisms using the monomorphism . The skew polynomial ring then corresponds to that of and the endormorphism ${\displaystyle R}$ ${\displaystyle R\ni a\mapsto (x\mapsto a\cdot x)\in \mathrm {End} (R^{(\mathbb {N} )})}$ ${\displaystyle \mathrm {End} (R^{(\mathbb {N} )})}$ ${\displaystyle R}$

${\displaystyle \delta =(a_{n})_{n\in \mathbb {N} }\mapsto {\bigl (}\,\sigma (a_{n})+\theta (a_{n-1})\,{\bigr )}_{n\in \mathbb {N} }\quad ({\text{wobei }}a_{-1}:=0)}$

generated subring of . More detailed explanations can be found in Chapter 0.10 in. ${\displaystyle \mathrm {End} (R^{(\mathbb {N} )})}$

Examples

• Ordinary polynomials are obtained by ( identity ) and . ${\displaystyle \sigma =\mathrm {id} }$ ${\displaystyle \theta =0}$
• At one speaks of differential operators . For example, the differential operators are infinitely often differentiable coefficients. ${\displaystyle \sigma =\mathrm {id} }$ ${\displaystyle C^{\infty }[\partial ;\mathrm {id} ,d/dx]}$
• The ring of shift operators with over polynomials with integer coefficients ${\displaystyle (\mathbb {Z} [t])[\partial ;\sigma ,0]}$ ${\displaystyle \sigma (f(t))=f(t+1)}$

properties

If zero divisors and injective , then applies ${\displaystyle R}$ ${\displaystyle \sigma }$

${\displaystyle \operatorname {grad} (ab)=\operatorname {grad} a+\operatorname {grad} b}$

for everyone . In particular, it is also free of zero divisors. ${\displaystyle a,b\in R[\partial ;\sigma ,\theta ]}$ ${\displaystyle R[\partial ;\sigma ,\theta ]}$

If the base ring is a body and an automorphism , then left and right-hand division with remainder can be defined. The greatest common right divider and greatest common left divider can then be calculated using a variant of the Euclidean algorithm . ${\displaystyle R}$ ${\displaystyle \sigma }$

Web links

• An introduction to pseudo-linear algebra
• OreTools ( skew polynomials in Maple )

swell

1. ↑ Öystein Ore [sic]: Formal theory of linear differential equations. (First part). In: Journal for pure and applied mathematics. Vol. 167, 1932, pp. 221-234, doi : 10.1515 / crll.1932.167.221 .
2. ^ Paul M. Cohn : Free Rings and their relations (= London Mathematical Society Monographs. 19). 2nd edition. London Academic Press, London et al. 1985, ISBN 0-12-179152-1 .
3. ↑ Manuel Bronstein, Marko Petkovšek: An introduction to pseudo-linear algebra. In: Theoretical Computer Science. Vol. 157, No. 1, 1996, pp. 3-33, doi : 10.1016 / 0304-3975 (95) 00173-5 .