ApCoCoA-1:CharP.GBasisF256: Difference between revisions
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<command> | <command> | ||
<title> | <title>CharP.GBasisF256</title> | ||
<short_description> | <short_description>Computing a Groebner Basis of a given ideal in <tt>F_256</tt>.</short_description> | ||
<syntax> | <syntax> | ||
CharP.GBasisF256(Ideal:IDEAL):LIST | |||
</syntax> | </syntax> | ||
<description> | <description> | ||
This command computes a Groebner basis in the field < | <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/> | |||
This command computes a Groebner basis in the field <tt>F_256 = (Z/(2))[x]/(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1)</tt>. | |||
<itemize> | |||
< | <item>@param <em>Ideal</em> An Ideal in a Ring over <tt>Z</tt>, where the elements <tt>0,...,255</tt> represent the elements of the field <tt>F_256</tt>. For short, the binary representation of the number represents the coefficient vector if 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> | ||
So the number < | <item>@return A Groebner Basis of the given ideal.</item> | ||
</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]; | |||
-- WARNING: Coeffs are not in a field | |||
-- GBasis-related computations could fail to terminate or be wrong | |||
------------------------------- | |||
I:=Ideal(x-y^2,x^2+xy,y^3); | |||
CharP.GBasisF256(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 + 144x, x^2, xy] | |||
------------------------------- | |||
</example> | |||
</description> | </description> | ||
<seealso> | <seealso> | ||
<see>GBasis</see> | <see>ApCoCoA-1:GBasis|GBasis</see> | ||
<see> | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | ||
<see> | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</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.GBasisF512|CharP.GBasisF512</see> | ||
<see> | <see>ApCoCoA-1:CharP.GBasisF1024|CharP.GBasisF1024</see> | ||
<see> | <see>ApCoCoA-1:CharP.GBasisF2048|CharP.GBasisF2048</see> | ||
<see>ApCoCoA-1:CharP.GBasisModSquares|CharP.GBasisModSquares</see> | |||
<see>ApCoCoA-1:Representation of finite fields|Representation of finite fields</see> | |||
</seealso> | </seealso> | ||
<key> | |||
<key> | <types> | ||
<wiki-category> | <type>apcocoaserver</type> | ||
<type>ideal</type> | |||
<type>groebner</type> | |||
</types> | |||
<key>GBasisF256</key> | |||
<key>charP.GBasisF256</key> | |||
<key>finite field</key> | |||
<wiki-category>ApCoCoA-1:Package_charP</wiki-category> | |||
</command> | </command> | ||
Latest revision as of 09:54, 7 October 2020
| This article is about a function from ApCoCoA-1. |
CharP.GBasisF256
Computing a Groebner Basis of a given ideal in F_256.
Syntax
CharP.GBasisF256(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_256 = (Z/(2))[x]/(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1).
@param Ideal An Ideal in a Ring over Z, where the elements 0,...,255 represent the elements of the field F_256. For short, the binary representation of the number represents the coefficient vector if 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]; -- WARNING: Coeffs are not in a field -- GBasis-related computations could fail to terminate or be wrong ------------------------------- I:=Ideal(x-y^2,x^2+xy,y^3); CharP.GBasisF256(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 + 144x, x^2, xy] -------------------------------
See also
Introduction to Groebner Basis in CoCoA
Representation of finite fields
