Difference between revisions of "ApCoCoA-1:CharP.GBasisF256"
| Line 1: | Line 1: | ||
<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 | ||
| Line 8: | Line 8: | ||
<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). | + | 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> | + | <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> | ||
| Line 53: | Line 53: | ||
<see>Char2.GBasisF1024</see> | <see>Char2.GBasisF1024</see> | ||
<see>Char2.GBasisF2048</see> | <see>Char2.GBasisF2048</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
Introduction to Groebner Basis in CoCoA
Representation of finite fields
