Difference between revisions of "ApCoCoA-1:CharP.GBasisF1024"
From ApCoCoAWiki
m (insert version info) |
|||
| (13 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| + | {{Version|1}} | ||
<command> | <command> | ||
| − | <title> | + | <title>CharP.GBasisF1024</title> |
| − | <short_description>Computing a Groebner basis of a given ideal in < | + | <short_description>Computing a Groebner basis of a given ideal in <tt>F_1024</tt>.</short_description> |
<syntax> | <syntax> | ||
| − | + | CharP.GBasisF1024(Ideal:IDEAL):LIST | |
</syntax> | </syntax> | ||
<description> | <description> | ||
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||
<par/> | <par/> | ||
| − | This command computes a Groebner basis in the field < | + | This command computes a Groebner basis in the field <tt>F_1024 = (Z/(2))[x]/(x^10 + x^3 + x^2 + x + 1)</tt>. |
<itemize> | <itemize> | ||
| − | <item>@param <em>Ideal</em> An Ideal in a Ring over Z, where the elements 0,...,1023 represent the elements of the finite field. For short, the binary representation of the number represents the coefficient vector of the polynomial in the field, e.g. 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0. So the number 11 corresponds to the polynomial x^3 + x + 1.</item> | + | <item>@param <em>Ideal</em> An Ideal in a Ring over <tt>Z</tt>, where the elements <tt>0,...,1023</tt> represent the elements of the finite field. For short, the binary representation of the number represents the coefficient vector of the polynomial in the field, e.g. <tt>11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0</tt>. So the number <tt>11</tt> corresponds to the polynomial <tt>x^3 + x + 1</tt>.</item> |
<item>@return A Groebner Basis of the given ideal.</item> | <item>@return A Groebner Basis of the given ideal.</item> | ||
</itemize> | </itemize> | ||
| + | |||
| + | <example> | ||
| + | Use R::=QQ[x,y,z]; | ||
| + | I:=Ideal(x-y^2,x^2+xy,y^3); | ||
| + | GBasis(I); | ||
| + | [x^2 + xy, -y^2 + x, -xy] | ||
| + | ------------------------------- | ||
| + | Use Z::=ZZ[x,y,z]; | ||
| + | I:=Ideal(x-y^2,x^2+xy,y^3); | ||
| + | -- WARNING: Coeffs are not in a field | ||
| + | -- GBasis-related computations could fail to terminate or be wrong | ||
| + | CharP.GBasisF1024(I); | ||
| + | ------------------------------- | ||
| + | -- WARNING: Coeffs are not in a field | ||
| + | -- GBasis-related computations could fail to terminate or be wrong | ||
| + | -- CoCoAServer: computing Cpu Time = 0 | ||
| + | ------------------------------- | ||
| + | [y^2 + 218x, x^2, xy] | ||
| + | ------------------------------- | ||
| + | </example> | ||
| + | |||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>GBasis</see> | + | <see>ApCoCoA-1:GBasis|GBasis</see> |
| − | <see>Introduction to Groebner Basis in CoCoA</see> | + | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see> |
| − | <see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF4|CharP.GBasisF4</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF8|CharP.GBasisF8</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF16|CharP.GBasisF16</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF32|CharP.GBasisF32</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF64|CharP.GBasisF64</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF128|CharP.GBasisF128</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF256|CharP.GBasisF256</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF512|CharP.GBasisF512</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF2048|CharP.GBasisF2048</see> |
| − | + | <see>ApCoCoA-1:CharP.GBasisModSquares|CharP.GBasisModSquares</see> | |
| − | <see>Representation of finite fields</see> | + | <see>ApCoCoA-1:Representation of finite fields|Representation of finite fields</see> |
</seealso> | </seealso> | ||
| Line 37: | Line 59: | ||
<type>groebner</type> | <type>groebner</type> | ||
<type>ideal</type> | <type>ideal</type> | ||
| − | <type> | + | <type>apcocoaserver</type> |
</types> | </types> | ||
| − | <key> | + | <key>charP.GBasisF1024</key> |
<key>GBasisF1024</key> | <key>GBasisF1024</key> | ||
| − | <wiki-category> | + | <key>finite field</key> |
| + | <wiki-category>ApCoCoA-1:Package_charP</wiki-category> | ||
</command> | </command> | ||
Latest revision as of 09:53, 7 October 2020
| This article is about a function from ApCoCoA-1. |
CharP.GBasisF1024
Computing a Groebner basis of a given ideal in F_1024.
Syntax
CharP.GBasisF1024(Ideal:IDEAL):LIST
Description
Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.
This command computes a Groebner basis in the field F_1024 = (Z/(2))[x]/(x^10 + x^3 + x^2 + x + 1).
@param Ideal An Ideal in a Ring over Z, where the elements 0,...,1023 represent the elements of the finite field. For short, the binary representation of the number represents the coefficient vector of the polynomial in the field, e.g. 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0. So the number 11 corresponds to the polynomial x^3 + x + 1.
@return A Groebner Basis of the given ideal.
Example
Use R::=QQ[x,y,z]; I:=Ideal(x-y^2,x^2+xy,y^3); GBasis(I); [x^2 + xy, -y^2 + x, -xy] ------------------------------- Use Z::=ZZ[x,y,z]; I:=Ideal(x-y^2,x^2+xy,y^3); -- WARNING: Coeffs are not in a field -- GBasis-related computations could fail to terminate or be wrong CharP.GBasisF1024(I); ------------------------------- -- WARNING: Coeffs are not in a field -- GBasis-related computations could fail to terminate or be wrong -- CoCoAServer: computing Cpu Time = 0 ------------------------------- [y^2 + 218x, x^2, xy] -------------------------------
See also
Introduction to Groebner Basis in CoCoA
Representation of finite fields
