ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 07 Aug 2019 18:14:35 -0500Define the affine variety $X = V (y − x^2, y − x + 1)$.http://ask.sagemath.org/question/47429/define-the-affine-variety-x-v-y-x2-y-x-1/Define the affine variety
(a) $X = V (y − x^2, y − x + 1)$.
(b) Find all the rational points on X.
What I got in the examples is that we can use the code
sage: x,y,z = PolynomialRing(GF(5), 3, 'xyz').gens()
sage: C = Curve(y^2*z^7 - x^9 - x*z^8); C
sage: C.rational_points()
To get rational points over Finite Field of size 5. To calculate over rational we can replace $GF(5)$ by QQ but to get a finite result we have to have the intersection.
Later I also used:
sage: R.<x,y> = PolynomialRing(QQ)
sage: R
sage: I = R.ideal(y-x^2,y-x+1)
sage: I.variety()
But didn't get my result.ArnabWed, 07 Aug 2019 18:14:35 -0500http://ask.sagemath.org/question/47429/sub-module membership testhttp://ask.sagemath.org/question/47428/sub-module-membership-test/I have a submodule of the module $\mathbb{Z}_2[x,y,z]^3$, which can be specified by its 6 generators that are the columns below
\begin{array}{cccccc}
0 & 0 & 0 & 1+x+y+xy & 1+y+z+yz & 1+x+z+xz\newline
1+z & 1+x & 0 & 0 & y+z & 0\newline
0 & 1+x& 1+y & x+y & 0 & z^2
\end{array}
How can I implement the sub-module membership test in sage? For example, I want to check whether
\begin{array}{c}
x+z \newline
x+y\newline
y+z
\end{array}
belongs to the above submodule or not?arpitWed, 07 Aug 2019 17:21:32 -0500http://ask.sagemath.org/question/47428/vector space basis for a quotient modulehttp://ask.sagemath.org/question/45173/vector-space-basis-for-a-quotient-module/For my question, let's say I have the following quotient module as an example,
\begin{align*}
& \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\left(\begin{array}{cccccc}
0 & 0 & 0 & 1+x+y+xy & 1+y+z+yz & 1+x+z+xz\newline
1+z & 1+x & 0 & 0 & 0 & 0\newline
0 & 1+x& 1+y & 0 & 0 & 0
\end{array}\right)}
\end{align*}
where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{Z}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.arpitFri, 25 Jan 2019 00:08:46 -0600http://ask.sagemath.org/question/45173/ideal membership and solutionhttp://ask.sagemath.org/question/45932/ideal-membership-and-solution/I gave sage the following ring and the ideal
R.<x,y,z>=GF(2)[];
f=1 + z + y*z + y^2*z + z^2 + y*z^2;
g=1 + x + y^2 + z^2;
I = R.ideal(f, g)
I found that the function h below lies in the Ideal I using
h=1 + y + z + x*z + y*z + x*y*z + y^2*z + y*z^2;
h in I
I know that in general, finding polynomials $a(x)$ and $b(x)$ such that $h = a f+ b g$ might be hard, but can I find the solutions for $a$ and $b$ to a certain degree of these polynomials, if they exist? I was wondering if sage can check this more efficiently
arpitThu, 28 Mar 2019 00:29:57 -0500http://ask.sagemath.org/question/45932/Ideals and commutative ringshttp://ask.sagemath.org/question/37359/ideals-and-commutative-rings/ I consider a matrix $M$ which I transform into a system of equations similar as this (but with a different $M$)
sage: M=matrix(3,3,[1,2,3,4,5,6,7,8,9])
sage: P=PolynomialRing(GF(p),M.nrows(),names="x")
sage: (vector(P.gen(i) for i in range(3))*M).list()
[x0 + 4*x1 + 7*x2, 2*x0 + 5*x1 + 8*x2, 3*x0 + 6*x1 + 9*x2]
(Example taken from another equation where $p$ is a prime).
When I try to create the ideal generated by this system, I get the following error : TypeError: R must be a commutative ring. Any idea how I can fix this?
Sasha-dptFri, 21 Apr 2017 07:18:51 -0500http://ask.sagemath.org/question/37359/how can I Calculate grobner bases in this application?http://ask.sagemath.org/question/36407/how-can-i-calculate-grobner-bases-in-this-application/ I want to solve some questions ,which they need to find many Grobner Bases but I don't know how can I use this app to find Grobner Bases , I will be so Thanks full if some one helps me .mahyaSat, 28 Jan 2017 10:09:33 -0600http://ask.sagemath.org/question/36407/