Twisted polynomial ring

In mathematics, a twisted polynomial is a polynomial over a field of characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} in the variable $\tau$ representing the Frobenius map $x\mapsto x^{p}$. In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies the commutation rule

$\tau x=x^{p}\tau$

for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the base field.

Over an infinite field, the twisted polynomial ring is isomorphic to the ring of additive polynomials, but where multiplication on the latter is given by composition rather than usual multiplication. However, it is often easier to compute in the twisted polynomial ring — this can be applied especially in the theory of Drinfeld modules.

Definition

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} be a field of characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . The twisted polynomial ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\{\tau\}} is defined as the set of polynomials in the variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} and coefficients in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau x=x^p\tau} for $x\in k$. Repeated application of this relation yields a formula for the multiplication of any two twisted polynomials.

As an example we perform such a multiplication

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+b\tau)(c+d\tau)=a(c+d\tau)+b\tau(c+d\tau)=ac+ad\tau+bc^p\tau+bd^p\tau^2}

Properties

The morphism

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\{\tau\}\to k[x],\quad a_0+a_1\tau+\cdots+a_n\tau^n\mapsto a_0x+a_1x^p+\cdots+a_nx^{p^n}}

defines a ring homomorphism sending a twisted polynomial to an additive polynomial. Here, multiplication on the right hand side is given by composition of polynomials. For example

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ax+bx^p)\circ (cx+dx^p)=a(cx+dx^p)+b(cx+dx^p)^p=acx+adx^p+bc^px^p+bd^px^{p^2},}

using the fact that in characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} we have the Freshman's dream Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x+y)^p=x^p+y^p} .

The homomorphism is clearly injective, but is surjective if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is infinite. The failure of surjectivity when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is finite is due to the existence of non-zero polynomials which induce the zero function on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} (e.g. $x^{q}-x$ over the finite field with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} elements).

Even though this ring is not commutative, it still possesses (left and right) division algorithms.

References

• Goss, D. (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131, Zbl 0874.11004
• Rosen, Michael (2002), Number Theory in Function Fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, ISBN 0-387-95335-3, ISSN 0072-5285, Zbl 1043.11079