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2022-04-11 05:56:31 +0100 | edited question | issue with sagemath on windows 11 issue with sagemath on windows 11 I installed sagemath on windows but get the following on running bash.exe: warning: |
2022-04-11 05:56:08 +0100 | asked a question | issue with sagemath on windows 11 issue with sagemath on windows 11 I installed sagemath on windows but get the following bash.exe: warning: could not f |
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2020-04-13 00:57:46 +0100 | asked a question | Finding polynomial solutions that belong to an ideal I have a system of equations in which the variables belong to a certain ideal of a polynomial ring over a field. We can call this ideal $I$ and its generators $c_1$, $c_2$ and $c_3$. Let's take the ring to be $\mathbb{F}_2[x_1,x_2,x_3]$ and the ideal to be $I=<x_1x_2x_3-1,x_2-x_1,x_1-1>$ . Let's say the equations are $g_1+x_1w_2+x_2w_3 = x_1-1$, $g_2+x_2w_1+x_3w_3 = x_2-1$ and $g_3+x_1w_1+w_2 = x_3-1$ and one needs to find solution to the above set of equations with variables $g_i$ and $w_i$ inside the ideal $I$. There are of course more variables than equations here. One obvious solution is $g_1=x_1-1,g_2=x_2-1, g_3=x_3-1$. How does one find the full set of solutions? I thought of implementing this as a syzygy problem where I take $x_1-1$ and so on on the left but that seems to be not ideal since it is not clear whether I find all solutions or not. To simplify the problem, we can choose a cut-off for degree of polynomial solutions, for example, 1 or 2. |
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2020-04-04 19:23:39 +0100 | asked a question | a basis for quotient module/vector space I have a ring (field) $R$, a polynomial ring $R[x_1,x_2,...,x_n]$ and a quotient module (vector space) $R[x_1,x_2,...,x_n]/I$ where $I$ is an ideal of $R[x_1,x_2,...,x_n]$ . For the case, when $R$ is a field, the basis of the quotient vector space can be found to consist of the cosets of monomials which are not divisible by leading monomials of grobner basis of $I$. A similar theorem exists for the case when $R$ is a ring with effective coset representatives. For example, the ring of integers. Can I find these coset representatives that form the basis for the quotient module/vector space in sage? |
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2020-02-18 19:30:48 +0100 | asked a question | intersection of two free submodules I have two modules found as follows [ 0 0 1] [y + 1 z + 1 0] [ 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). |
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2019-08-10 05:49:27 +0100 | commented question | sub-module membership test over the polynomial ring $\mathbb{Z}_2[x,y,z]$ |
2019-08-08 00:21:32 +0100 | asked a question | 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? |
2019-08-07 23:53:57 +0100 | edited question | 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. |
2019-04-10 06:43:21 +0100 | commented answer | parametric solution for a system of polynomial equations @slelievre, thanks for the answer. But I want to also know how to apply it more generally i.e. can I also find the variety in the cases when I don't know the form of the parametrization? for example, if the polynomials are |