I need to implement the following formal structure $$ax^\gamma, \gamma \in \mathbb{Z}/n\mathbb{Z}, a \in \mathbb{R}$$ and $x$ is a formal variable.
I tried
ZZ6 = Integers(6)
x = var('x')
x^ZZ6(9) + x^ZZ6(11)
And get value x^8, whereas I need x^ZZ6(8).
How can I do this in sage?
You can make a polynomial quotient ring:
sage: R.<x> = RR[]
sage: R
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: S = R.quotient(x^6-1)
sage: S
Univariate Quotient Polynomial Ring in xbar over Real Field with 53 bits of precision with modulus x^6 - 1.00000000000000
sage: S.inject_variables()
Defining xbar
Then, you can do
sage: xbar^9 + xbar^11
xbar^5 + xbar^3
sage: xbar^9 * xbar^11
xbar^2
Also, if your coefficients turn out to be rational, i would sugest to define sage: R.<x> = QQ[]
This solution has the drawback of introducing an inexact ring in otherwise exact computations...
@emmanuel-charpentier : i used RR to stay close to @only1sale question, I then explained that QQ could be used otherwise, i do not get your point.
another way
sage: R = Zmod(6)
sage: A = R.algebra(QQ,category=CommutativeAdditiveSemigroups())
sage: x = A.gens()[1]
sage: 3*x+x**4
3*B[1] + B[4]
sage: _**4
120*B[1] + 136*B[4]
Asked: 2020-06-09 11:15:48 +0100
Seen: 361 times
Last updated: Jun 09 '20
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.