Difference between revisions of "ApCoCoA-1:CharP.GBasisF2048"

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(Added ApCoCoAServer note)
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{{Version|1}}
 
<command>
 
<command>
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     <title>Char2.GBasisF2048</title>
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     <title>CharP.GBasisF2048</title>
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     <short_description>computing a gbasis of a given ideal in <formula>\mathbb{F}_{2048}</formula></short_description>
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     <short_description>Computing a Groebner Basis of a given ideal in <tt>F_2048</tt>.</short_description>
 
<syntax>
 
<syntax>
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$char2.GBasisF2048(Ideal):List
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CharP.GBasisF2048(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_2048 = (Z/(2))[x]/(x^11 +x^3 + x^5 +x + 1)</tt>. 
  
−
This command computes a Groebner basis in the field <formula> \mathbb{F}_{2048} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^{11} +x^3 + x^5 +x + 1)}</formula>. It uses the ApCoCoA Server and the ApCoCoALib's class [[ApCoCoALib:RingF2048|RingF2048]].  
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<itemize>
 +
<item>@param <em>Ideal</em> An Ideal in a Ring over <tt>Z</tt>, where the elements <tt>0,...,2047</tt> represent the elements of the field <tt>F_2048</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>
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<item>@return A Groebner Basis of the given ideal.</item>
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</itemize>
  
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The command's input is a an Ideal in a Ring over Z, where the elements 0,..., 2047 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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<example>
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<formula> 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0</formula>
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Use R::=QQ[x,y,z];
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So the number <formula>11</formula> corresponds to the polynomial <formula>x^3 + x + 1</formula>.
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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];
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-- 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.GBasisF2048(I);
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-- WARNING: Coeffs are not in a field
 +
-- GBasis-related computations could fail to terminate or be wrong
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 +
[y^2 + 1205x, 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 CoCoAServer|Introduction to CoCoAServer</see>
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     <see>char2.GBasisF4</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.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.GBasisF1024</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.GBasisF1024|CharP.GBasisF1024</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.GBasisF2048</key>
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    <types>
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     <wiki-category>Package_char2</wiki-category>
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      <type>apcocoaserver</type>
 +
      <type>ideal</type>
 +
      <type>groebner</type>
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    </types>
 +
 
 +
     <key>charP.GBasisF2048</key>
 +
     <key>GBasisF2048</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.GBasisF2048

Computing a Groebner Basis of a given ideal in F_2048.

Syntax

CharP.GBasisF2048(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_2048 = (Z/(2))[x]/(x^11 +x^3 + x^5 +x + 1).

  • @param Ideal An Ideal in a Ring over Z, where the elements 0,...,2047 represent the elements of the field F_2048. 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.GBasisF2048(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 + 1205x, x^2, xy]
-------------------------------


See also

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.GBasisF4

CharP.GBasisF8

CharP.GBasisF16

CharP.GBasisF32

CharP.GBasisF64

CharP.GBasisF128

CharP.GBasisF256

CharP.GBasisF512

CharP.GBasisF1024

CharP.GBasisModSquares

Representation of finite fields





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