/// <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);
}
/// <summary>
/// Computes the Groebner basis for a polynomial ideal using Buchbergers algorithm.
/// </summary>
/// <param name="basis">The basis that defines the ideal.</param>
/// <returns>The Groebner basis.</returns>
public static PolynomialBasis GetGroebnerBasis(params Polynomial[] basis)
{
PolynomialBasis groebner = new PolynomialBasis(basis);
return(groebner.SimplifiedBuchberger());
//return groebner.OptimizedBuchberger(basis); //TODO: Change to optimized Buchberger.
}
/// <summary>
/// Implementation incomplete.
/// </summary>
/// <param name="basis"></param>
/// <returns></returns>
public static PolynomialBasis GrobnerBasisOptimized(params Polynomial[] basis)
{
int t = basis.Length;
List <Tuple <int, int> > b = new List <Tuple <int, int> >();
for (int i = 0; i < t; i++)
{
for (int j = i; j < t; j++)
{
b.Add(Tuple.Create(i, j));
}
}
PolynomialBasis groebner = new PolynomialBasis(basis);
while (b.Count > 0)
{
int i = b[0].Item1;
int j = b[0].Item2;
/*
* if (basis[i].GetLeadingTerm().LCM(basis[j].GetLeadingTerm()).Equals(basis[i].GetLeadingTerm()) && !criterion()) // TODO: Implement multiply monomials.
* {
*
* }*/
}
return(groebner);
}
/// <summary>
/// Gets remainder on division by a polynomial basis.
/// </summary>
/// <param name="basis">The basis (or collection) of quitient polynomials.</param>
/// <returns>The remainder polynomial on division.</returns>
public Polynomial GetRemainder(PolynomialBasis basis)
{
Polynomial[] divisorList = basis.polynomialData.ToArray();
List <Polynomial> quotients = this.DivideBy(divisorList);
return(quotients[quotients.Count - 1]); // The last element in the list is the remainder
}
public static void CLOSecn2Pt7Exerc2a()
{
Monomial.orderingScheme = "lex";
Polynomial p1 = new Polynomial(new double[] { 1, -1 }, new Monomial(2, 1), new Monomial(0, 0));
Polynomial p2 = new Polynomial(new double[] { 1, -1 }, new Monomial(1, 2), new Monomial(1, 0));
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(p1, p2);
PrintPolynomialBasis(gb);
}
/// <summary>
/// Initializes a new instance of class <see cref="PolynomialBasis"/>. Creates a deep copy of the input basis in a new instance.
/// </summary>
/// <param name="inputBasis">The input basis which is to be deep copied.</param>
public PolynomialBasis(PolynomialBasis inputBasis)
{
this.polynomialData = new HashSet <Polynomial>();
foreach (Polynomial p in inputBasis.polynomialData)
{
Polynomial toAdd = new Polynomial(p);
this.polynomialData.Add(toAdd);
}
}
public static void CLOSectn2Pt8Exerc11()
{
Monomial.orderingScheme = "lex";
Polynomial p1 = new Polynomial(new double[] { 1, 1, 1, -3 }, new Monomial(1, 0, 0), new Monomial(0, 1, 0), new Monomial(0, 0, 1), new Monomial(0, 0, 0));
Polynomial p2 = new Polynomial(new double[] { 1, 1, 1, -5 }, new Monomial(2, 0, 0), new Monomial(0, 2, 0), new Monomial(0, 0, 2), new Monomial(0, 0, 0));
Polynomial p3 = new Polynomial(new double[] { 1, 1, 1, -7 }, new Monomial(3, 0, 0), new Monomial(0, 3, 0), new Monomial(0, 0, 3), new Monomial(0, 0, 0));
Polynomial p4 = new Polynomial(new double[] { 1, 1, 1, -9 }, new Monomial(4, 0, 0), new Monomial(0, 4, 0), new Monomial(0, 0, 4), new Monomial(0, 0, 0));
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(p1, p2, p3, p4);
PrintPolynomialBasis(gb);
}
public static void CLOSec2Pt7Exerc9()
{
Monomial.orderingScheme = "lex";
Polynomial p1 = new Polynomial(new double[] { 3, -6, -2 }, new Monomial(1, 0, 0, 0), new Monomial(0, 1, 0, 0), new Monomial(0, 0, 1, 0));
Polynomial p2 = new Polynomial(new double[] { 2, -4, 4 }, new Monomial(1, 0, 0, 0), new Monomial(0, 1, 0, 0), new Monomial(0, 0, 0, 1));
Polynomial p3 = new Polynomial(new double[] { 1, -2, -1, -1 }, new Monomial(1, 0, 0, 0), new Monomial(0, 1, 0, 0), new Monomial(0, 0, 1, 0), new Monomial(0, 0, 0, 1));
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(p1, p2, p3);
PrintPolynomialBasis(gb);
}
/// <summary>
/// Optimized Buchberger implementation based on Theorem 11 of section 2.9.
/// </summary>
/// <param name="basis">An array of polynomials that are going to form the basis.</param>
/// <returns>A polynomial basis that wraps the polynomials that make up the Groebner basis.</returns>
public PolynomialBasis OptimizedBuchberger(params Polynomial[] basis)
{
int t = basis.Length - 1;
HashSet <Tuple <int, int> > b = new HashSet <Tuple <int, int> >();
for (int i = 0; i <= t; i++)
{
for (int j = i + 1; j <= t; j++)
{
b.Add(Tuple.Create(i, j)); // i will always be less than j.
}
}
PolynomialBasis groebner = new PolynomialBasis(basis);
List <Polynomial> listBasis = basis.ToList();
while (b.Count > 0)
{
List <Tuple <int, int> > bList = b.ToList();
foreach (Tuple <int, int> ij in bList)
{
int i = ij.Item1;
int j = ij.Item2;
if ( // First, is the combination relatively prime (see propositions 4 of section 2.9, CLO)? If so, we know the remainder by G will be zero.
// So, just skip the if condition and remove it.
listBasis[i].GetLeadingTerm().LCM(listBasis[j].GetLeadingTerm()) != listBasis[i].GetLeadingTerm() * (listBasis[j].GetLeadingTerm())
// Next, is it possible to remove this combination? If so, skip the if and remove ij.
&& !this.Criterion(i, j, b, listBasis)
)
{
Polynomial s = listBasis[i].GetSPolynomial(listBasis[j]).GetRemainder(new PolynomialBasis(listBasis)); // TODO: Implement remainder method that takes list. It will be more efficient.
if (!s.IsZero)
{
t++;
listBasis.Add(s);
for (int l = 0; l < t; l++)
{ // We need to consider all combinations with this new polynomial that was introduced.
b.Add(Tuple.Create(l, t)); // l will always be smaller than t.
}
}
}
b.Remove(ij); // ij has already been considered and its S-polynomial (if non-zero) added. So, we don't need to consider it again.
}
}
PolynomialBasis grb = new PolynomialBasis(listBasis);
grb.MakeMinimal(); // First minimal and then reduced.
grb.MakeReduced();
return(grb);
}
public static void CLOSecn2Pt8Exerc5()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"4x^3+4xy^2-8x-3",
"4y^3+4yx^2-8y-3"
);
PolynomialBasis gb = pb.SimplifiedBuchberger();
gb.PrettyPrint();
}
public static void TestOptimizedBuchberger()
{
Monomial.orderingScheme = "grlex";
PolynomialBasis pb = new PolynomialBasis
(
"x^3 - 2xy",
"x^2y - 2y^2 + x"
);
PolynomialBasis gb = pb.OptimizedBuchberger(); // since gb doesn't have pb's indiceToVariable, we will get old fashioned printing.
gb.PrettyPrint();
}
public static void PrintPolynomialBasis(PolynomialBasis gb)
{
int i = 0;
foreach (Polynomial p in gb.polynomialData)
{
System.Console.WriteLine("Polynomial - " + i++.ToString());
foreach (Monomial m in p.monomialData.Keys)
{
System.Console.WriteLine(string.Join(",", m.powers) + " * " + p.monomialData[m].ToString() + " + ");
}
}
}
/**************************
* Now for expercies from section 8 of CLO
*************************/
public static void CLOSecn2Pt8Exerc1()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"xy^3 - z^2 + y^5 - z^3",
"-x^3 + y",
"x^2y - z"
);
PolynomialBasis gb = pb.OptimizedBuchberger();
gb.PrettyPrint();
}
public static void CLOSecn2Pt8Exerc3()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"x^2 + y^2 + z^2 - 1", // Ensure that y appears before z in the equations.
"x^2 + y^2 + z^2 - 2x",
"2x - 3y - z"
);
PolynomialBasis gb = pb.SimplifiedBuchberger();
gb.PrettyPrint();
}
public static void CLOSecn2Pt8Exerc6()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"t + u -x",
"t^2 + 2tu - y",
"t^3+3t^2u-z"
);
PolynomialBasis gb = pb.OptimizedBuchberger();
gb.PrettyPrint();
}
public static void TestString2GroebnerBasis()
{
Monomial.orderingScheme = "grlex";
PolynomialBasis pb = new PolynomialBasis
(
"x^2 + y^2 + z^2 - 8",
"x^2 + z^2 - y",
"x - z + y - 2"
);
pb.SimplifiedBuchberger();
pb.PrettyPrint();
}
public static void CLOSecn2Pt8Exerc7()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"ut - x",
"1-u-y",
"u+t-ut-z"
);
PolynomialBasis gb = pb.SimplifiedBuchberger();
gb.PrettyPrint();
}
public static void TestGroebnerBasisCLOSec2Pt7Exerc3()
{
Monomial.orderingScheme = "lex";
Polynomial f = new Polynomial(new Monomial(new int[] { 2, 1 }));
f.AddMonomial(new Monomial(new int[] { 0, 0 }), -1);
Polynomial g = new Polynomial(new Monomial(new int[] { 1, 2 }));
g.AddMonomial(new Monomial(new int[] { 1, 0 }), -1);
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(f, g);
PrintPolynomialBasis(gb);
}
public static void CLOSecn2Pt8Exerc2()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"xy + 2z^2", // Ensure that y appears before z in the equations.
"y - z",
"xz - y",
"x^3z - 2y^2"
);
PolynomialBasis gb = pb.SimplifiedBuchberger();
gb.PrettyPrint();
}
public static void CLOSecn2Pt8Exerc9()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"a^2 + b^2 -1",
"8a^5 -2a^3 -3a - x",
"8b^5 - 18b^3 + 9b -y",
"2ab - z"
);
PolynomialBasis gb = pb.OptimizedBuchberger();
gb.PrettyPrint();
}
public static void CLOSecn2Pt8Exerc4()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"x^2y - z^3",
"2xy - 4z - 1",
"z - y^2",
"x^3 - 4zy"
);
PolynomialBasis gb = pb.OptimizedBuchberger();
gb.PrettyPrint();
}
public static void CLOSecn2Pt8Exerc10()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"4lx + 2x - 2",
"2ly + 2y - 2",
"2lz + 2z - 2",
"x^4 + y^2 + z^2 -1"
);
PolynomialBasis gb = pb.OptimizedBuchberger();
gb.PrettyPrint();
}
public static void TestGroebnerBasisCLOSecn2Pt7Example1()
{
Monomial.orderingScheme = "grlex";
Polynomial f = new Polynomial(new Monomial(new int[] { 3, 0 })); // x^3
f.AddMonomial(new Monomial(new int[] { 1, 1 }), -2.0); // -2xy
Polynomial g = new Polynomial(new Monomial(new int[] { 2, 1 })); // x^2y
g.AddMonomial(new Monomial(new int[] { 0, 2 }), -2.0); // -2y^2
g.AddMonomial(new Monomial(new int[] { 1, 0 })); // x
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(f, g);
PrintPolynomialBasis(gb);
}
public static void CLOSecn2Pt8Exerc8()
{
Monomial.orderingScheme = "lex";
PolynomialBasis pb = new PolynomialBasis
(
"a^2 + b^2 -1",
"c^2 + d^2 -1",
"ac + 2c - x",
"ad + 2d - y",
"b - z"
);
PolynomialBasis gb = pb.SimplifiedBuchberger();
gb.PrettyPrint();
}
public static void TestGroebnerBasisLinearSystem()
{
Monomial.orderingScheme = "lex";
Polynomial f = new Polynomial(new Monomial(new int[] { 1, 0 }));
f.AddMonomial(new Monomial(new int[] { 0, 1 }), 2);
f.AddMonomial(new Monomial(new int[] { 0, 0 }), -3);
Polynomial g = new Polynomial(new Monomial(new int[] { 1, 0 }));
g.AddMonomial(new Monomial(new int[] { 0, 1 }), -2);
g.AddMonomial(new Monomial(new int[] { 0, 0 }), -7);
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(f, g);
PrintPolynomialBasis(gb);
}
/// <summary>
/// Checks to see if the two polynomial bases are equal.
/// </summary>
/// <param name="obj">An object that is parsed to another polynomial with which equality is to be tested.</param>
/// <returns>A boolean value specifying if this polynomial basis is equal to the other or not.</returns>
public override bool Equals(object obj)
{
PolynomialBasis other = (PolynomialBasis)obj;
if (this.polynomialData.Count != other.polynomialData.Count)
{
return(false);
}
foreach (Polynomial p in other.polynomialData)
{
if (!this.polynomialData.Contains(p))
{
return(false);
}
}
return(true);
}
/// <summary>
/// Test method for Groebner basis.
/// </summary>
public static void TestGroebnerBasis()
{
Monomial.orderingScheme = "lex";
//1st order
var del_x = 1.0;
var del_y = 2.0;
//2nd order
var del_x_x = 3.5;
var del_x_y = 1.5;
var del_y_y = 2.7;
//3rd order
var del_y_y_y = 1.1;
var del_x_y_y = 1.9;
var del_x_x_y = 2.3;
var del_x_x_x = 1.4;
Polynomial f = new Polynomial(new Monomial(new int[] { 2, 0 }), del_x_x_x);
f.AddMonomial(new Monomial(new int[] { 1, 1 }), 2 * del_x_x_y);
f.AddMonomial(new Monomial(new int[] { 0, 2 }), del_x_y_y);
f.AddMonomial(new Monomial(new int[] { 1, 0 }), del_x_y);
f.AddMonomial(new Monomial(new int[] { 0, 1 }), del_y_y);
f.AddMonomial(new Monomial(new int[] { 0, 0 }), -del_x);
Polynomial g = new Polynomial(new Monomial(new int[] { 2, 0 }), del_x_x_y);
g.AddMonomial(new Monomial(new int[] { 1, 1 }), 2 * del_x_y_y);
g.AddMonomial(new Monomial(new int[] { 0, 2 }), del_y_y_y);
g.AddMonomial(new Monomial(new int[] { 1, 0 }), del_x_x);
g.AddMonomial(new Monomial(new int[] { 0, 1 }), del_x_y);
g.AddMonomial(new Monomial(new int[] { 0, 0 }), -del_y);
PolynomialBasis gb = PolynomialBasis.GetGroebnerBasis(f, g);
PrintPolynomialBasis(gb);
}
/// <summary>
/// Computes the Groebner basis for a polynomial ideal using Buchbergers algorithm.
/// </summary>
/// <param name="basis">The basis that defines the ideal.</param>
/// <returns>The Groebner basis.</returns>
public static PolynomialBasis GroebnerBasis(params Polynomial[] basis)
{
PolynomialBasis groebner = new PolynomialBasis(basis);
return(groebner.SimplifiedGroebnerBasis());
}