Difference between revisions of "ApCoCoA-1:CharP.GBasisF16"

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m (insert version info)
 
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   <command>
+
   {{Version|1}}
     <title>Char2.GBasisF16</title>
+
<command>
     <short_description>computing a gbasis of a given ideal in <formula>\mathbb{F}_16</formula</short_description>
+
     <title>CharP.GBasisF16</title>
 +
     <short_description>Computing a Groebner Basis of a given ideal in <tt>F_16</tt>.</short_description>
 
<syntax>
 
<syntax>
$char2.GBasisF16(Ideal):List
+
CharP.GBasisF16(Ideal):List
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
This function returns reduced Groebner basis for the ideal, intersected with the ideal, created by <formula>x^2-x</formula> for all indeterminates. If <formula>x^2-x</formula> for
+
<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.
all indeterminates is in the ideal (e.g. the set of zeros is a subset of <formula>\{0,1\}^n</formula>) this method should produce the GBasis much faster!
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<par/>
Please be aware, that this is much more efficient if the term ordering is Lex, DegLex or DegRevLex. Otherwise, first a DegRevLex GBasis is computed and then
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This command computes a Groebner basis in the field <tt>F_16 = (Z/(2))[x]/(x^4 + x^3 +1)</tt>.
transformed with the FGLM-algorithm.
+
 
 +
<itemize>
 +
<item>@param <em>Ideal</em> An Ideal in a Ring over <tt>Z</tt>, where the elements <tt>0,...,15</tt> represent the elements of the field <tt>F_16</tt>. 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>
 +
<item>@return The 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.GBasisF16(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 + 8x, x^2, xy]
 +
-------------------------------
 +
</example>
 +
 
 
     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
       <see>FGLM</see>
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       <see>ApCoCoA-1:GBasis|GBasis</see>
      <see>GBasis</see>
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    <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
     </seealso>
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    <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
     <key>heldt</key>
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    <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see>
     <key>char2.gbasismodsquares</key>
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    <see>ApCoCoA-1:CharP.GBasisF4|CharP.GBasisF4</see>
     <wiki-category>Package_char2</wiki-category>
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    <see>ApCoCoA-1:CharP.GBasisF8|CharP.GBasisF8</see>
 +
    <see>ApCoCoA-1:CharP.GBasisF32|CharP.GBasisF32</see>
 +
    <see>ApCoCoA-1:CharP.GBasisF64|CharP.GBasisF64</see>
 +
    <see>ApCoCoA-1:CharP.GBasisF128|CharP.GBasisF128</see>
 +
    <see>ApCoCoA-1:CharP.GBasisF256|CharP.GBasisF256</see>
 +
    <see>ApCoCoA-1:CharP.GBasisF512|CharP.GBasisF512</see>
 +
    <see>ApCoCoA-1:CharP.GBasisF1024|CharP.GBasisF1024</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>
 +
     <types>
 +
      <type>apcocoaserver</type>
 +
      <type>ideal</type>
 +
      <type>groebner</type>
 +
    </types>
 +
    <key>charP.GBasisF16</key>
 +
     <key>GBasisF16</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.GBasisF16

Computing a Groebner Basis of a given ideal in F_16.

Syntax

CharP.GBasisF16(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_16 = (Z/(2))[x]/(x^4 + x^3 +1).

  • @param Ideal An Ideal in a Ring over Z, where the elements 0,...,15 represent the elements of the field F_16. 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 The 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.GBasisF16(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 + 8x, x^2, xy]
-------------------------------


See also

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.GBasisF4

CharP.GBasisF8

CharP.GBasisF32

CharP.GBasisF64

CharP.GBasisF128

CharP.GBasisF256

CharP.GBasisF512

CharP.GBasisF1024

CharP.GBasisF2048

CharP.GBasisModSquares

Representation of finite fields