Difference between revisions of "ApCoCoA-1:CharP.GBasisF256"
| Line 11: | Line 11: | ||
<itemize> | <itemize> | ||
| − | <item>@param <em>Ideal</em> An Ideal in a Ring over Z, where the elements 0,...,255 represent the field | + | <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>@return A Groebner Basis of the given ideal.</item> | <item>@return A 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); | ||
| + | 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] | ||
| + | ------------------------------- | ||
| + | </example> | ||
| + | |||
</description> | </description> | ||
Revision as of 07:38, 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
