Difference between revisions of "ApCoCoA-1:CharP.GBasisF16"
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| − | <command> | + | {{Version|1}} |
| − | <title> | + | <command> |
| − | <short_description>Computing a Groebner Basis of a given ideal in < | + | <title>CharP.GBasisF16</title> |
| + | <short_description>Computing a Groebner Basis of a given ideal in <tt>F_16</tt>.</short_description> | ||
<syntax> | <syntax> | ||
| − | + | CharP.GBasisF16(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. | <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_16 = (Z/(2))[x]/(x^4 + x^3 +1). | + | This command computes a Groebner basis in the field <tt>F_16 = (Z/(2))[x]/(x^4 + x^3 +1)</tt>. |
<itemize> | <itemize> | ||
| − | <item>@param <em>Ideal</em> An Ideal in a Ring over Z, where the elements 0,..., 15 represent the field | + | <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> | <item>@return The Groebner Basis of the given ideal.</item> | ||
</itemize> | </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>GBasis</see> | + | <see>ApCoCoA-1:GBasis|GBasis</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
| − | <see>Introduction to Groebner Basis in CoCoA</see> | + | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF4|CharP.GBasisF4</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF8|CharP.GBasisF8</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF32|CharP.GBasisF32</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF64|CharP.GBasisF64</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF128|CharP.GBasisF128</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF256|CharP.GBasisF256</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF512|CharP.GBasisF512</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF1024|CharP.GBasisF1024</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisF2048|CharP.GBasisF2048</see> |
| − | <see> | + | <see>ApCoCoA-1:CharP.GBasisModSquares|CharP.GBasisModSquares</see> |
| − | + | <see>ApCoCoA-1:Representation of finite fields|Representation of finite fields</see> | |
| − | <see>Representation of finite fields</see> | ||
</seealso> | </seealso> | ||
<types> | <types> | ||
| Line 39: | Line 62: | ||
<type>groebner</type> | <type>groebner</type> | ||
</types> | </types> | ||
| − | <key> | + | <key>charP.GBasisF16</key> |
<key>GBasisF16</key> | <key>GBasisF16</key> | ||
| − | <wiki-category> | + | <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
Introduction to Groebner Basis in CoCoA
Representation of finite fields
