Difference between revisions of "ApCoCoA-1:CharP.GBasisF1024"

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(Added ApCoCoAServer note)
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{{Version|1}}
 
<command>
 
<command>
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     <title>Char2.GBasisF1024</title>
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     <title>CharP.GBasisF1024</title>
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     <short_description>computing a gbasis of a given ideal in <formula>\mathbb{F}_{1024}</formula></short_description>
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     <short_description>Computing a Groebner basis of a given ideal in <tt>F_1024</tt>.</short_description>
 
<syntax>
 
<syntax>
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$char2.GBasisF1024(Ideal):List
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CharP.GBasisF1024(Ideal:IDEAL):LIST
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
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{{ApCoCoAServer}}
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<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_1024 = (Z/(2))[x]/(x^10 + x^3 + x^2 + x + 1)</tt>.
  
−
This command computes a Groebner basis in the field <formula> \mathbb{F}_{1024} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^{10} + x^3 + x^2 + x + 1)}</formula>. It uses the ApCoCoA Server and the ApCoCoALib's class [[ApCoCoALib:RingF1024|RingF1024]].
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<itemize>
 +
<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>
  
−
The command's input is a an Ideal in a Ring over Z, where the elements 0,..., 1023 represent the field's elements. Details on this representation can be found [[ApCoCoA:Representation_of_finite_fields|here]]. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g.
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<item>@return A Groebner Basis of the given ideal.</item>
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<formula> 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0</formula>
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</itemize>
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So the number <formula>11</formula> corresponds to the polynomial <formula>x^3 + x + 1</formula>.
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 +
<example>
 +
Use R::=QQ[x,y,z];
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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>
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       <see>GBasis</see>
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       <see>ApCoCoA-1:GBasis|GBasis</see>
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     <see>char2.GBasisF2</see>  
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    <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
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     <see>char2.GBasisF4</see>
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     <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
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     <see>char2.GBasisF8</see>
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    <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see>  
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     <see>char2.GBasisF16</see>
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     <see>ApCoCoA-1:CharP.GBasisF4|CharP.GBasisF4</see>
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     <see>char2.GBasisF32</see>
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     <see>ApCoCoA-1:CharP.GBasisF8|CharP.GBasisF8</see>
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     <see>char2.GBasisF64</see>
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     <see>ApCoCoA-1:CharP.GBasisF16|CharP.GBasisF16</see>
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     <see>char2.GBasisF128</see>
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     <see>ApCoCoA-1:CharP.GBasisF32|CharP.GBasisF32</see>
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     <see>char2.GBasisF256</see>
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     <see>ApCoCoA-1:CharP.GBasisF64|CharP.GBasisF64</see>
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     <see>char2.GBasisF512</see>
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     <see>ApCoCoA-1:CharP.GBasisF128|CharP.GBasisF128</see>
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     <see>char2.GBasisF2048</see>
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     <see>ApCoCoA-1:CharP.GBasisF256|CharP.GBasisF256</see>
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     <see>char2.GBasisF4096</see>
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     <see>ApCoCoA-1:CharP.GBasisF512|CharP.GBasisF512</see>
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     <see>char2.GBasisModSquares</see>
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     <see>ApCoCoA-1:CharP.GBasisF2048|CharP.GBasisF2048</see>
−
 
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     <see>ApCoCoA-1:CharP.GBasisModSquares|CharP.GBasisModSquares</see>
 +
     <see>ApCoCoA-1:Representation of finite fields|Representation of finite fields</see>
 
   </seealso>
 
   </seealso>
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     <key>heldt</key>
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−
     <key>char2.GBasisF1024</key>
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<types>
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     <wiki-category>Package_char2</wiki-category>
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<type>groebner</type>
 +
<type>ideal</type>
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<type>apcocoaserver</type>
 +
</types>
 +
 
 +
     <key>charP.GBasisF1024</key>
 +
     <key>GBasisF1024</key>
 +
    <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

GBasis

Introduction to Groebner Basis in CoCoA

Introduction to CoCoAServer

CharP.GBasisF2

CharP.GBasisF4

CharP.GBasisF8

CharP.GBasisF16

CharP.GBasisF32

CharP.GBasisF64

CharP.GBasisF128

CharP.GBasisF256

CharP.GBasisF512

CharP.GBasisF2048

CharP.GBasisModSquares

Representation of finite fields





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