ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 18 Feb 2020 19:30:48 +0100intersection of two free submoduleshttps://ask.sagemath.org/question/49966/intersection-of-two-free-submodules/ I have two modules found as follows
F=GF(2);R.<x,y,z> = PolynomialRing(F)
f1 = 1+z;g1=1+y;h1=0;
I1 = Ideal([f1,g1,h1])
M1 = I1.syzygy_module(); M1
[ 0 0 1]
[y + 1 z + 1 0]
F=GF(2);R.<x,y,z> = PolynomialRing(F)
f2 = 0;g2=1+y;h2=1+x;
I2 = Ideal([f2,g2,h2])
M2 = I2.syzygy_module(); M2
[ 1 0 0]
[ 0 x + 1 y + 1]
Is it possible to find the intersection of two such submodules $M_1$ and $M_2$ in sage? Another possibility would be to find the syzygy of the module generated by vectors (f1,g1,h1) and (f2,g2,h2). arpitTue, 18 Feb 2020 19:30:48 +0100https://ask.sagemath.org/question/49966/sub-module membership testhttps://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?arpitThu, 08 Aug 2019 00:21:32 +0200https://ask.sagemath.org/question/47428/vector space basis for a quotient modulehttps://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 07:08:46 +0100https://ask.sagemath.org/question/45173/ideal membership and solutionhttps://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 06:29:57 +0100https://ask.sagemath.org/question/45932/