Difference between revisions of "ApCoCoA-1:CharP.GBasisF32"

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<command>
 
<command>
     <title>Char2.GBasisF32</title>
+
     <title>CharP.GBasisF32</title>
     <short_description>computing a gbasis of a given ideal in <formula>\mathbb{F}_{32}</formula></short_description>
+
     <short_description>Computing a Groebner Basis of a given ideal in <tt>F_32</tt>.</short_description>
 
<syntax>
 
<syntax>
$char2.GBasisF32(Ideal):List
+
CharP.GBasisF32(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.
 +
<par/>
 +
This command computes a Groebner basis in the field <tt>F_32 = (Z/(2))[x]/(x^5 + x^2 + 1)</tt>.
 +
 +
<itemize>
 +
<item>@param <em>Ideal</em> An Ideal in a Ring over <tt>Z</tt>, where the elements <tt>0,...,31</tt> represent the elements of the field <tt>F_32</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>
 +
</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.GBasisF32(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 + 27x, x^2, xy]
 +
-------------------------------
 +
</example>
  
 
     </description>
 
     </description>
 +
 
     <seealso>
 
     <seealso>
 
       <see>GBasis</see>
 
       <see>GBasis</see>
     <see>char2.GBasisF2</see>
+
     <see>Introduction to CoCoAServer</see>
     <see>char2.GBasisF4</see>
+
    <see>Introduction to Groebner Basis in CoCoA</see>
     <see>char2.GBasisF8</see>
+
    <see>CharP.GBasisF2</see>
     <see>char2.GBasisF16</see>
+
     <see>CharP.GBasisF4</see>
     <see>char2.GBasisF64</see>
+
     <see>CharP.GBasisF8</see>
     <see>char2.GBasisF128</see>
+
     <see>CharP.GBasisF16</see>
     <see>char2.GBasisF256</see>
+
     <see>CharP.GBasisF64</see>
     <see>char2.GBasisF512</see>
+
     <see>CharP.GBasisF128</see>
     <see>char2.GBasisF1024</see>
+
     <see>CharP.GBasisF256</see>
     <see>char2.GBasisF2048</see>
+
     <see>CharP.GBasisF512</see>
     <see>char2.GBasisF4096</see>
+
     <see>CharP.GBasisF1024</see>
     <see>char2.GBasisModSquares</see>
+
     <see>CharP.GBasisF2048</see>
 
+
     <see>CharP.GBasisModSquares</see>
 +
     <see>Representation of finite fields</see>
 
   </seealso>
 
   </seealso>
     <key>heldt</key>
+
 
     <key>char2.GBasisF32</key>
+
    <types>
     <wiki-category>Package_char2</wiki-category>
+
      <type>apcocoaserver</type>
 +
      <type>ideal</type>
 +
      <type>groebner</type>
 +
    </types>
 +
 
 +
     <key>GBasisF32</key>
 +
     <key>charP.GBasisF32</key>
 +
    <key>finite field</key>
 +
     <wiki-category>Package_charP</wiki-category>
 
   </command>
 
   </command>

Revision as of 15:13, 6 December 2010

CharP.GBasisF32

Computing a Groebner Basis of a given ideal in F_32.

Syntax

CharP.GBasisF32(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_32 = (Z/(2))[x]/(x^5 + x^2 + 1).

  • @param Ideal An Ideal in a Ring over Z, where the elements 0,...,31 represent the elements of the field F_32. 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.GBasisF32(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 + 27x, x^2, xy]
-------------------------------


See also

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.GBasisF4

CharP.GBasisF8

CharP.GBasisF16

CharP.GBasisF64

CharP.GBasisF128

CharP.GBasisF256

CharP.GBasisF512

CharP.GBasisF1024

CharP.GBasisF2048

CharP.GBasisModSquares

Representation of finite fields