/*
 * $Id$
 */

package edu.jas.gb;

import java.util.List;
import java.io.Serializable;

import edu.jas.poly.GenSolvablePolynomial;

import edu.jas.structure.RingElem;


/**
 * Solvable polynomial Reduction interface.
 * Defines S-Polynomial, normalform
 * and irreducible set.
 * @param  coefficient type
 * @author Heinz Kredel
 */

public interface SolvableReduction> extends Serializable {


    /**
     * Left S-Polynomial.
     * @param Ap solvable polynomial.
     * @param Bp solvable polynomial.
     * @return left-spol(Ap,Bp) the left S-polynomial of Ap and Bp.
     */
    public GenSolvablePolynomial 
           leftSPolynomial(GenSolvablePolynomial Ap, 
                           GenSolvablePolynomial Bp);


    /**
     * S-Polynomial with recording.
     * @param S recording matrix, is modified.
     * @param i index of Ap in basis list.
     * @param Ap a polynomial.
     * @param j index of Bp in basis list.
     * @param Bp a polynomial.
     * @return leftSpol(Ap, Bp), the left S-Polynomial for Ap and Bp.
     */
    public GenSolvablePolynomial 
           leftSPolynomial(List> S,
                           int i,
                           GenSolvablePolynomial Ap, 
                           int j,
                           GenSolvablePolynomial Bp);


    /**
     * Left Normalform.
     * @param Ap solvable polynomial.
     * @param Pp solvable polynomial list.
     * @return left-nf(Ap) with respect to Pp.
     */
    public GenSolvablePolynomial 
           leftNormalform(List> Pp, 
                          GenSolvablePolynomial Ap);


    /**
     * LeftNormalform with recording.
     * @param row recording matrix, is modified.
     * @param Pp a polynomial list for reduction.
     * @param Ap a polynomial.
     * @return nf(Pp,Ap), the left normal form of Ap wrt. Pp.
     */
    public GenSolvablePolynomial 
           leftNormalform(List> row,
                          List> Pp, 
                          GenSolvablePolynomial Ap);


    /**
     * Left Normalform Set.
     * @param Ap solvable polynomial list.
     * @param Pp solvable polynomial list.
     * @return list of left-nf(a) with respect to Pp for all a in Ap.
     */
    public List> 
           leftNormalform(List> Pp, 
                          List> Ap);


    /**
     * Left irreducible set.
     * @param Pp solvable polynomial list.
     * @return a list P of solvable polynomials which are in normalform wrt. P.
     */
    public List> 
           leftIrreducibleSet(List> Pp); 


    /**
     * Is reduction of normal form.
     * @param row recording matrix, is modified.
     * @param Pp a solvable polynomial list for reduction.
     * @param Ap a solvable polynomial.
     * @param Np nf(Pp,Ap), a left normal form of Ap wrt. Pp.
     * @return true, if Np + sum( row[i]*Pp[i] ) == Ap, else false.
     */

    public boolean 
           isLeftReductionNF(List> row,
                             List> Pp, 
                             GenSolvablePolynomial Ap,
                             GenSolvablePolynomial Np);


    /**
     * Right S-Polynomial.
     * @param Ap solvable polynomial.
     * @param Bp solvable polynomial.
     * @return right-spol(Ap,Bp) the right S-polynomial of Ap and Bp.
     */
    public GenSolvablePolynomial 
           rightSPolynomial(GenSolvablePolynomial Ap, 
                            GenSolvablePolynomial Bp);


    /**
     * Right Normalform.
     * @param Ap solvable polynomial.
     * @param Pp solvable polynomial list.
     * @return right-nf(Ap) with respect to Pp.
     */
    public GenSolvablePolynomial 
           rightNormalform(List> Pp, 
                           GenSolvablePolynomial Ap);

    /**
     * RightNormalform with recording.
     * @param row recording matrix, is modified.
     * @param Pp a polynomial list for reduction.
     * @param Ap a polynomial.
     * @return nf(Pp,Ap), the right normal form of Ap wrt. Pp.
     */
    public GenSolvablePolynomial 
           rightNormalform(List> row,
                           List> Pp, 
                           GenSolvablePolynomial Ap);

}