/// <summary>
/// Computes Groebner basis for the polynomial basis. Based on Theorem 2 of sectin 2.7, CLO. Not very efficient.
/// </summary>
/// <returns>A polynomial basis that is the reduced Groebner basis.</returns>
public PolynomialBasis SimplifiedBuchberger()
{
PolynomialBasis groebner = this;
PolynomialBasis groebnerTemp = this;
do
{
groebnerTemp = new PolynomialBasis(groebner); // G' := G
List <Polynomial> groebnerTempList = groebnerTemp.polynomialData.ToList();
for (int i = 0; i < groebnerTemp.polynomialData.Count; i++)
{
for (int j = (i + 1); j < groebnerTemp.polynomialData.Count; j++)
{
Polynomial s = groebnerTempList[i].GetSPolynomial(groebnerTempList[j]).GetRemainder(groebnerTemp);
if (!s.IsZero)
{
groebner.polynomialData.Add(s); // G := G ∪ {S}
}
}
}
}while (!groebner.Equals(groebnerTemp));
groebner.MakeMinimal(); // First minimal and then reduced.
groebner.MakeReduced();
return(groebner);
}
