Difference between revisions of "ApCoCoA-1:CharP.GBasisF256"

From ApCoCoAWiki
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<command>
 
<command>
 
     <title>Char2.GBasisF256</title>
 
     <title>Char2.GBasisF256</title>
     <short_description>Computing a Groebner Basis of a given ideal in F_256.</short_description>
+
     <short_description>Computing a Groebner Basis of a given ideal in <tt>F_256</tt>.</short_description>
 
<syntax>
 
<syntax>
 
Char2.GBasisF256(Ideal:IDEAL):LIST
 
Char2.GBasisF256(Ideal:IDEAL):LIST
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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.
 
<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 F_256 = (Z/(2))[x]/(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1).  
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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>
 
<itemize>
<item>@param <em>Ideal</em> 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.</item>
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<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>
 
<item>@return A Groebner Basis of the given ideal.</item>
 
<item>@return A Groebner Basis of the given ideal.</item>
 
</itemize>
 
</itemize>
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     <see>Char2.GBasisF1024</see>
 
     <see>Char2.GBasisF1024</see>
 
     <see>Char2.GBasisF2048</see>
 
     <see>Char2.GBasisF2048</see>
    <see>Char2.GBasisF4096</see>
 
 
     <see>Char2.GBasisModSquares</see>
 
     <see>Char2.GBasisModSquares</see>
 
     <see>Representation of finite fields</see>
 
     <see>Representation of finite fields</see>

Revision as of 08:26, 14 July 2009

Char2.GBasisF256

Computing a Groebner Basis of a given ideal in F_256.

Syntax

Char2.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);
Char2.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

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

Char2.GBasisF2

Char2.GBasisF4

Char2.GBasisF8

Char2.GBasisF16

Char2.GBasisF32

Char2.GBasisF64

Char2.GBasisF128

Char2.GBasisF512

Char2.GBasisF1024

Char2.GBasisF2048

Char2.GBasisModSquares

Representation of finite fields