Processing math: 100%

First time here? Check out the FAQ!

Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

asked 6 years ago

arpit gravatar image

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(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]3 is a polynomial ring in variables x,y,z over field F2. 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.

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(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]3 is a polynomial ring in variables x,y,z over field F2. 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. great.

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(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]3 is a polynomial ring in variables x,y,z over field F2. 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.

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} \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{6}}{\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 Z2[x,y,z]3 Z2[x,y,z]6 is a polynomial ring in variables x,y,z over field F2. 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.

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)^{6}}{\left(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]6 is a polynomial ring in variables x,y,z over field F2. Z2. 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.

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)^{6}}{\left(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]6 is a polynomial ring in variables x,y,z over field Z2. 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.

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)^{6}}{\left(\begin{array}{cccccc} \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 Z2[x,y,z]6 Z2[x,y,z]3 is a polynomial ring in variables x,y,z over field Z2. 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.

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(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]3 is a polynomial ring in variables x,y,z over field Z2. 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.

click to hide/show revision 9
retagged

updated 5 years ago

FrédéricC gravatar image

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(0001+x+y+xy1+y+z+yz1+x+z+xz1+z1+x000001+x1+y000\right)} \end{align}

where Z2[x,y,z]3 is a polynomial ring in variables x,y,z over field Z2. 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.