Laboratoire de l'Informatique et du Parallélisme » Pobyso

r"""

Sage core functions needed for the implementation of SLZ.

AUTHORS:

- S.T. (2013-08): initial version

Examples:

TODO::

"""

sys.stderr.write("sageSLZ loading...\n")

#

import inspect

#

def slz_compute_binade(number):

    """"

    For a given number, compute the "binade" that is integer m such that

    2^m <= number < 2^(m+1). If number == 0 return None.

    """

    # Checking the parameter.

    # The exception construction is used to detect if number is a RealNumber

    # since not all numbers have

    # the mro() method. sage.rings.real_mpfr.RealNumber do.

    try:

        classTree = [number.__class__] + number.mro()

        # If the number is not a RealNumber (or offspring thereof) try

        # to transform it.

        if not sage.rings.real_mpfr.RealNumber in classTree:

            numberAsRR = RR(number)

        else:

            numberAsRR = number

    except AttributeError:

        return None

    # Zero special case.

    if numberAsRR == 0:

        return RR(-infinity)

    else:

        realField           = numberAsRR.parent()

        numberLog2          = numberAsRR.abs().log2()

        floorNumberLog2     = floor(numberLog2)

        ## Do not get caught by rounding of log2() both ways.

        ## When numberLog2 is an integer, compare numberAsRR

        #  with 2^numberLog2.

        if floorNumberLog2 == numberLog2:

            if numberAsRR.abs() < realField(2^floorNumberLog2):

                return floorNumberLog2 - 1

            else:

                return floorNumberLog2

        else:

            return floorNumberLog2

# End slz_compute_binade

#

def slz_compute_binade_bounds(number, emin, emax=sys.maxint):

    """

    For given "real number", compute the bounds of the binade it belongs to.

    NOTE::

        When number >= 2^(emax+1), we return the "fake" binade 

        [2^(emax+1), +infinity]. Ditto for number <= -2^(emax+1)

        with interval [-infinity, -2^(emax+1)]. We want to distinguish

        this case from that of "really" invalid arguments.

    """

    # Check the parameters.

    # RealNumbers or RealNumber offspring only.

    # The exception construction is necessary since not all objects have

    # the mro() method. sage.rings.real_mpfr.RealNumber do.

    try:

        classTree = [number.__class__] + number.mro()

        if not sage.rings.real_mpfr.RealNumber in classTree:

            return None

    except AttributeError:

        return None

    # Non zero negative integers only for emin.

    if emin >= 0 or int(emin) != emin:

        return None

    # Non zero positive integers only for emax.

    if emax <= 0 or int(emax) != emax:

        return None

    precision = number.precision()

    RF  = RealField(precision)

    if number == 0:

        return (RF(0),RF(2^(emin)) - RF(2^(emin-precision)))

    # A more precise RealField is needed to avoid unwanted rounding effects

    # when computing number.log2().

    RRF = RealField(max(2048, 2 * precision))

    # number = 0 special case, the binade bounds are 

    # [0, 2^emin - 2^(emin-precision)]

    # Begin general case

    l2 = RRF(number).abs().log2()

    # Another special one: beyond largest representable -> "Fake" binade.

    if l2 >= emax + 1:

        if number > 0:

            return (RF(2^(emax+1)), RF(+infinity) )

        else:

            return (RF(-infinity), -RF(2^(emax+1)))

    # Regular case cont'd.

    offset = int(l2)

    # number.abs() >= 1.

    if l2 >= 0:

        if number >= 0:

            lb = RF(2^offset)

            ub = RF(2^(offset + 1) - 2^(-precision+offset+1))

        else: #number < 0

            lb = -RF(2^(offset + 1) - 2^(-precision+offset+1))

            ub = -RF(2^offset)

    else: # log2 < 0, number.abs() < 1.

        if l2 < emin: # Denormal

           # print "Denormal:", l2

            if number >= 0:

                lb = RF(0)

                ub = RF(2^(emin)) - RF(2^(emin-precision))

            else: # number <= 0

                lb = - RF(2^(emin)) + RF(2^(emin-precision))

                ub = RF(0)

        elif l2 > emin: # Normal number other than +/-2^emin.

            if number >= 0:

                if int(l2) == l2:

                    lb = RF(2^(offset))

                    ub = RF(2^(offset+1)) - RF(2^(-precision+offset+1))

                else:

                    lb = RF(2^(offset-1))

                    ub = RF(2^(offset)) - RF(2^(-precision+offset))

            else: # number < 0

                if int(l2) == l2: # Binade limit.

                    lb = -RF(2^(offset+1) - 2^(-precision+offset+1))

                    ub = -RF(2^(offset))

                else:

                    lb = -RF(2^(offset) - 2^(-precision+offset))

                    ub = -RF(2^(offset-1))

        else: # l2== emin, number == +/-2^emin

            if number >= 0:

                lb = RF(2^(offset))

                ub = RF(2^(offset+1)) - RF(2^(-precision+offset+1))

            else: # number < 0

                lb = -RF(2^(offset+1) - 2^(-precision+offset+1))

                ub = -RF(2^(offset))

    return (lb, ub)

# End slz_compute_binade_bounds

#

def slz_compute_coppersmith_reduced_polynomials(inputPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound,

                                                 debug = False):

    """

    For a given set of arguments (see below), compute a list

    of "reduced polynomials" that could be used to compute roots

    of the inputPolynomial.

    INPUT:

    - "inputPolynomial" -- (no default) a bivariate integer polynomial;

    - "alpha" -- the alpha parameter of the Coppersmith algorithm;

    - "N" -- the modulus;

    - "iBound" -- the bound on the first variable;

    - "tBound" -- the bound on the second variable.

    OUTPUT:

    A list of bivariate integer polynomial obtained using the Coppersmith

    algorithm. The polynomials correspond to the rows of the LLL-reduce

    reduced base that comply with the Coppersmith condition.

    """

    # Arguments check.

    if iBound == 0 or tBound == 0:

        return None

    # End arguments check.

    nAtAlpha = N^alpha

    ## Building polynomials for matrix.

    polyRing   = inputPolynomial.parent()

    # Whatever the 2 variables are actually called, we call them

    # 'i' and 't' in all the variable names.

    (iVariable, tVariable) = inputPolynomial.variables()[:2]

    #print polyVars[0], type(polyVars[0])

    initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,

                                              tVariable:tVariable * tBound})

    if debug:

        polynomialsList = \

            spo_polynomial_to_polynomials_list_8(initialPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound,

                                                 20)

    else:

        polynomialsList = \

            spo_polynomial_to_polynomials_list_8(initialPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound,

                                                 0)

    #print "Polynomials list:", polynomialsList

    ## Building the proto matrix.

    knownMonomials = []

    protoMatrix    = []

    if debug:

        for poly in polynomialsList:

            spo_add_polynomial_coeffs_to_matrix_row(poly,

                                                    knownMonomials,

                                                    protoMatrix,

                                                    20)

    else:

        for poly in polynomialsList:

            spo_add_polynomial_coeffs_to_matrix_row(poly,

                                                    knownMonomials,

                                                    protoMatrix,

                                                    0)

    matrixToReduce = spo_proto_to_row_matrix(protoMatrix)

    #print matrixToReduce

    ## Reduction and checking.

    ## S.T. changed 'fp' to None as of Sage 6.6 complying to

    #  error message issued when previous code was used.

    #reducedMatrix = matrixToReduce.LLL(fp='fp')

    reducedMatrix = matrixToReduce.LLL(fp=None)

    isLLLReduced  = reducedMatrix.is_LLL_reduced()

    if not isLLLReduced:

        return None

    monomialsCount     = len(knownMonomials)

    monomialsCountSqrt = sqrt(monomialsCount)

    #print "Monomials count:", monomialsCount, monomialsCountSqrt.n()

    #print reducedMatrix

    ## Check the Coppersmith condition for each row and build the reduced 

    #  polynomials.

    ccReducedPolynomialsList = []

    for row in reducedMatrix.rows():

        l2Norm = row.norm(2)

        if (l2Norm * monomialsCountSqrt) < nAtAlpha:

            #print (l2Norm * monomialsCountSqrt).n()

            #print l2Norm.n()

            ccReducedPolynomial = \

                slz_compute_reduced_polynomial(row,

                                               knownMonomials,

                                               iVariable,

                                               iBound,

                                               tVariable,

                                               tBound)

            if not ccReducedPolynomial is None:

                ccReducedPolynomialsList.append(ccReducedPolynomial)

        else:

            #print l2Norm.n() , ">", nAtAlpha

            pass

    if len(ccReducedPolynomialsList) < 2:

        print "Less than 2 Coppersmith condition compliant vectors."

        return ()

    #print ccReducedPolynomialsList

    return ccReducedPolynomialsList

# End slz_compute_coppersmith_reduced_polynomials

#

def slz_compute_coppersmith_reduced_polynomials_gram(inputPolynomial,

                                                     alpha,

                                                     N,

                                                     iBound,

                                                     tBound,

                                                     debug = False):

    """

    For a given set of arguments (see below), compute a list

    of "reduced polynomials" that could be used to compute roots

    of the inputPolynomial.

    INPUT:

    - "inputPolynomial" -- (no default) a bivariate integer polynomial;

    - "alpha" -- the alpha parameter of the Coppersmith algorithm;

    - "N" -- the modulus;

    - "iBound" -- the bound on the first variable;

    - "tBound" -- the bound on the second variable.

    OUTPUT:

    A list of bivariate integer polynomial obtained using the Coppersmith

    algorithm. The polynomials correspond to the rows of the LLL-reduce

    reduced base that comply with the Coppersmith condition.

    """

    # Arguments check.

    if iBound == 0 or tBound == 0:

        return None

    # End arguments check.

    nAtAlpha = N^alpha

    ## Building polynomials for matrix.

    polyRing   = inputPolynomial.parent()

    # Whatever the 2 variables are actually called, we call them

    # 'i' and 't' in all the variable names.

    (iVariable, tVariable) = inputPolynomial.variables()[:2]

    #print polyVars[0], type(polyVars[0])

    initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,

                                              tVariable:tVariable * tBound})

    if debug:

        polynomialsList = \

            spo_polynomial_to_polynomials_list_8(initialPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound,

                                                 20)

    else:

        polynomialsList = \

            spo_polynomial_to_polynomials_list_8(initialPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound,

                                                 0)

    #print "Polynomials list:", polynomialsList

    ## Building the proto matrix.

    knownMonomials = []

    protoMatrix    = []

    if debug:

        for poly in polynomialsList:

            spo_add_polynomial_coeffs_to_matrix_row(poly,

                                                    knownMonomials,

                                                    protoMatrix,

                                                    20)

    else:

        for poly in polynomialsList:

            spo_add_polynomial_coeffs_to_matrix_row(poly,

                                                    knownMonomials,

                                                    protoMatrix,

                                                    0)

    matrixToReduce = spo_proto_to_row_matrix(protoMatrix)

    #print matrixToReduce

    ## Reduction and checking.

    ### In this variant we use the Pari LLL_gram reduction function.

    #   It works with the Gram matrix instead of the plain matrix.

    matrixToReduceTransposed = matrixToReduce.transpose() 

    matrixToReduceGram = matrixToReduce * matrixToReduceTransposed

    ### LLL_gram returns a unimodular transformation matrix such that:

    #   umt.transpose() * matrixToReduce * umt is reduced..

    umt = matrixToReduceGram.LLL_gram()

    #print "Unimodular transformation matrix:"

    #for row in umt.rows():

    #    print row

    ### The computed transformation matrix is transposed and applied to the

    #   left side of matrixToReduce. 

    reducedMatrix = umt.transpose() * matrixToReduce

    #print "Reduced matrix:"

    #for row in reducedMatrix.rows():

    #    print row

    isLLLReduced  = reducedMatrix.is_LLL_reduced()

    #if not isLLLReduced:

    #    return None

    monomialsCount     = len(knownMonomials)

    monomialsCountSqrt = sqrt(monomialsCount)

    #print "Monomials count:", monomialsCount, monomialsCountSqrt.n()

    #print reducedMatrix

    ## Check the Coppersmith condition for each row and build the reduced 

    #  polynomials.

    ccReducedPolynomialsList = []

    for row in reducedMatrix.rows():

        l2Norm = row.norm(2)

        if (l2Norm * monomialsCountSqrt) < nAtAlpha:

            #print (l2Norm * monomialsCountSqrt).n()

            #print l2Norm.n()

            ccReducedPolynomial = \

                slz_compute_reduced_polynomial(row,

                                               knownMonomials,

                                               iVariable,

                                               iBound,

                                               tVariable,

                                               tBound)

            if not ccReducedPolynomial is None:

                ccReducedPolynomialsList.append(ccReducedPolynomial)

        else:

            #print l2Norm.n() , ">", nAtAlpha

            pass

    if len(ccReducedPolynomialsList) < 2:

        print "Less than 2 Coppersmith condition compliant vectors."

        return ()

    #print ccReducedPolynomialsList

    return ccReducedPolynomialsList

# End slz_compute_coppersmith_reduced_polynomials_gram

#

def slz_compute_coppersmith_reduced_polynomials_with_lattice_volume(inputPolynomial,

                                                                    alpha,

                                                                    N,

                                                                    iBound,

                                                                    tBound,

                                                                    debug = False):

    """

    For a given set of arguments (see below), compute a list

    of "reduced polynomials" that could be used to compute roots

    of the inputPolynomial.

    Print the volume of the initial basis as well.

    INPUT:

    - "inputPolynomial" -- (no default) a bivariate integer polynomial;

    - "alpha" -- the alpha parameter of the Coppersmith algorithm;

    - "N" -- the modulus;

    - "iBound" -- the bound on the first variable;

    - "tBound" -- the bound on the second variable.

    OUTPUT:

    A list of bivariate integer polynomial obtained using the Coppersmith

    algorithm. The polynomials correspond to the rows of the LLL-reduce

    reduced base that comply with the Coppersmith condition.

    """

    # Arguments check.

    if iBound == 0 or tBound == 0:

        return None

    # End arguments check.

    nAtAlpha = N^alpha

    if debug:

        print "N at alpha:", nAtAlpha

    ## Building polynomials for matrix.

    polyRing   = inputPolynomial.parent()

    # Whatever the 2 variables are actually called, we call them

    # 'i' and 't' in all the variable names.

    (iVariable, tVariable) = inputPolynomial.variables()[:2]

    #print polyVars[0], type(polyVars[0])

    initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,

                                              tVariable:tVariable * tBound})

##    polynomialsList = \

##        spo_polynomial_to_polynomials_list_8(initialPolynomial,

##        spo_polynomial_to_polynomials_list_5(initialPolynomial,

    polynomialsList = \

        spo_polynomial_to_polynomials_list_5(initialPolynomial,

                                             alpha,

                                             N,

                                             iBound,

                                             tBound,

                                             0)

    #print "Polynomials list:", polynomialsList

    ## Building the proto matrix.

    knownMonomials = []

    protoMatrix    = []

    for poly in polynomialsList:

        spo_add_polynomial_coeffs_to_matrix_row(poly,

                                                knownMonomials,

                                                protoMatrix,

                                                0)

    matrixToReduce = spo_proto_to_row_matrix(protoMatrix)

    matrixToReduceTranspose = matrixToReduce.transpose()

    squareMatrix = matrixToReduce * matrixToReduceTranspose

    squareMatDet = det(squareMatrix)

    latticeVolume = sqrt(squareMatDet)

    print "Lattice volume:", latticeVolume.n()

    print "Lattice volume / N:", (latticeVolume/N).n()

    #print matrixToReduce

    ## Reduction and checking.

    ## S.T. changed 'fp' to None as of Sage 6.6 complying to

    #  error message issued when previous code was used.

    #reducedMatrix = matrixToReduce.LLL(fp='fp')

    reductionTimeStart = cputime() 

    reducedMatrix = matrixToReduce.LLL(fp=None)

    reductionTime = cputime(reductionTimeStart)

    print "Reduction time:", reductionTime

    isLLLReduced  = reducedMatrix.is_LLL_reduced()

    if not isLLLReduced:

       return None

    #

    if debug:

        matrixFile = file('/tmp/reducedMatrix.txt', 'w')

        for row in reducedMatrix.rows():

            matrixFile.write(str(row) + "\n")

        matrixFile.close()

    #

    monomialsCount     = len(knownMonomials)

    monomialsCountSqrt = sqrt(monomialsCount)

    #print "Monomials count:", monomialsCount, monomialsCountSqrt.n()

    #print reducedMatrix

    ## Check the Coppersmith condition for each row and build the reduced 

    #  polynomials.

    ccVectorsCount = 0

    ccReducedPolynomialsList = []

    for row in reducedMatrix.rows():

        l2Norm = row.norm(2)

        if (l2Norm * monomialsCountSqrt) < nAtAlpha:

            #print (l2Norm * monomialsCountSqrt).n()

            #print l2Norm.n()

            ccVectorsCount +=1            

            ccReducedPolynomial = \

                slz_compute_reduced_polynomial(row,

                                               knownMonomials,

                                               iVariable,

                                               iBound,

                                               tVariable,

                                               tBound)

            if not ccReducedPolynomial is None:

                ccReducedPolynomialsList.append(ccReducedPolynomial)

        else:

            #print l2Norm.n() , ">", nAtAlpha

            pass

    if debug:

        print ccVectorsCount, "out of ", len(ccReducedPolynomialsList), 

        print "took Coppersmith text."

    if len(ccReducedPolynomialsList) < 2:

        print "Less than 2 Coppersmith condition compliant vectors."

        return ()

    if debug:

        print "Reduced and Coppersmith compliant polynomials list", ccReducedPolynomialsList

    return ccReducedPolynomialsList

# End slz_compute_coppersmith_reduced_polynomials_with_lattice volume

def slz_compute_initial_lattice_matrix(inputPolynomial,

                                       alpha,

                                       N,

                                       iBound,

                                       tBound,

                                       debug = False):

    """

    For a given set of arguments (see below), compute the initial lattice

    that could be reduced. 

    INPUT:

    - "inputPolynomial" -- (no default) a bivariate integer polynomial;

    - "alpha" -- the alpha parameter of the Coppersmith algorithm;

    - "N" -- the modulus;

    - "iBound" -- the bound on the first variable;

    - "tBound" -- the bound on the second variable.

    OUTPUT:

    A list of bivariate integer polynomial obtained using the Coppersmith

    algorithm. The polynomials correspond to the rows of the LLL-reduce

    reduced base that comply with the Coppersmith condition.

    """

    # Arguments check.

    if iBound == 0 or tBound == 0:

        return None

    # End arguments check.

    nAtAlpha = N^alpha

    ## Building polynomials for matrix.

    polyRing   = inputPolynomial.parent()

    # Whatever the 2 variables are actually called, we call them

    # 'i' and 't' in all the variable names.

    (iVariable, tVariable) = inputPolynomial.variables()[:2]

    #print polyVars[0], type(polyVars[0])

    initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,

                                              tVariable:tVariable * tBound})

    polynomialsList = \

        spo_polynomial_to_polynomials_list_8(initialPolynomial,

                                             alpha,

                                             N,

                                             iBound,

                                             tBound,

                                             0)

    #print "Polynomials list:", polynomialsList

    ## Building the proto matrix.

    knownMonomials = []

    protoMatrix    = []

    for poly in polynomialsList:

        spo_add_polynomial_coeffs_to_matrix_row(poly,

                                                knownMonomials,

                                                protoMatrix,

                                                0)

    matrixToReduce = spo_proto_to_row_matrix(protoMatrix)

    if debug:

        print "Initial basis polynomials"

        for poly in polynomialsList:

            print poly

    return matrixToReduce

# End slz_compute_initial_lattice_matrix.

def slz_compute_integer_polynomial_modular_roots(inputPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound):

    """

    For a given set of arguments (see below), compute the polynomial modular 

    roots, if any.

    """

    # Arguments check.

    if iBound == 0 or tBound == 0:

        return set()

    # End arguments check.

    nAtAlpha = N^alpha

    ## Building polynomials for matrix.

    polyRing   = inputPolynomial.parent()

    # Whatever the 2 variables are actually called, we call them

    # 'i' and 't' in all the variable names.

    (iVariable, tVariable) = inputPolynomial.variables()[:2]

    ccReducedPolynomialsList = \

        slz_compute_coppersmith_reduced_polynomials (inputPolynomial,

                                                     alpha,

                                                     N,

                                                     iBound,

                                                     tBound)

    if len(ccReducedPolynomialsList) == 0:

        return set()   

    ## Create the valid (poly1 and poly2 are algebraically independent) 

    # resultant tuples (poly1, poly2, resultant(poly1, poly2)).

    # Try to mix and match all the polynomial pairs built from the 

    # ccReducedPolynomialsList to obtain non zero resultants.

    resultantsInITuplesList = []

    for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList)-1):

        for polyInnerIndex in xrange(polyOuterIndex+1, 

                                     len(ccReducedPolynomialsList)):

            # Compute the resultant in resultants in the

            # first variable (is it the optimal choice?).

            resultantInI = \

               ccReducedPolynomialsList[polyOuterIndex].resultant(ccReducedPolynomialsList[polyInnerIndex],

                                                                  ccReducedPolynomialsList[0].parent(str(iVariable)))

            #print "Resultant", resultantInI

            # Test algebraic independence.

            if not resultantInI.is_zero():

                resultantsInITuplesList.append((ccReducedPolynomialsList[polyOuterIndex],

                                                 ccReducedPolynomialsList[polyInnerIndex],

                                                 resultantInI))

    # If no non zero resultant was found: we can't get no algebraically 

    # independent polynomials pair. Give up!

    if len(resultantsInITuplesList) == 0:

        return set()

    #print resultantsInITuplesList

    # Compute the roots.

    Zi = ZZ[str(iVariable)]

    Zt = ZZ[str(tVariable)]

    polynomialRootsSet = set()

    # First, solve in the second variable since resultants are in the first

    # variable.

    for resultantInITuple in resultantsInITuplesList:

        tRootsList = Zt(resultantInITuple[2]).roots()

        # For each tRoot, compute the corresponding iRoots and check 

        # them in the input polynomial.

        for tRoot in tRootsList:

            #print "tRoot:", tRoot

            # Roots returned by root() are (value, multiplicity) tuples.

            iRootsList = \

                Zi(resultantInITuple[0].subs({resultantInITuple[0].variables()[1]:tRoot[0]})).roots()

            print iRootsList

            # The iRootsList can be empty, hence the test.

            if len(iRootsList) != 0:

                for iRoot in iRootsList:

                    polyEvalModN = inputPolynomial(iRoot[0], tRoot[0]) / N

                    # polyEvalModN must be an integer.

                    if polyEvalModN == int(polyEvalModN):

                        polynomialRootsSet.add((iRoot[0],tRoot[0]))

    return polynomialRootsSet

# End slz_compute_integer_polynomial_modular_roots.

#

def slz_compute_integer_polynomial_modular_roots_2(inputPolynomial,

                                                 alpha,

                                                 N,

                                                 iBound,

                                                 tBound):

    """

    For a given set of arguments (see below), compute the polynomial modular 

    roots, if any.

    This version differs in the way resultants are computed.   

    """

    # Arguments check.

    if iBound == 0 or tBound == 0:

        return set()

    # End arguments check.

    nAtAlpha = N^alpha

    ## Building polynomials for matrix.

    polyRing   = inputPolynomial.parent()

    # Whatever the 2 variables are actually called, we call them

    # 'i' and 't' in all the variable names.

    (iVariable, tVariable) = inputPolynomial.variables()[:2]

    #print polyVars[0], type(polyVars[0])

    ccReducedPolynomialsList = \

        slz_compute_coppersmith_reduced_polynomials (inputPolynomial,

                                                     alpha,

                                                     N,

                                                     iBound,

                                                     tBound)

    if len(ccReducedPolynomialsList) == 0:

        return set()   

    ## Create the valid (poly1 and poly2 are algebraically independent) 

    # resultant tuples (poly1, poly2, resultant(poly1, poly2)).

    # Try to mix and match all the polynomial pairs built from the 

    # ccReducedPolynomialsList to obtain non zero resultants.

    resultantsInTTuplesList = []

    for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList)-1):

        for polyInnerIndex in xrange(polyOuterIndex+1, 

                                     len(ccReducedPolynomialsList)):

            # Compute the resultant in resultants in the

            # first variable (is it the optimal choice?).

            resultantInT = \

               ccReducedPolynomialsList[polyOuterIndex].resultant(ccReducedPolynomialsList[polyInnerIndex],

                                                                  ccReducedPolynomialsList[0].parent(str(tVariable)))

            #print "Resultant", resultantInT

            # Test algebraic independence.

            if not resultantInT.is_zero():

                resultantsInTTuplesList.append((ccReducedPolynomialsList[polyOuterIndex],

                                                 ccReducedPolynomialsList[polyInnerIndex],

                                                 resultantInT))

    # If no non zero resultant was found: we can't get no algebraically 

    # independent polynomials pair. Give up!

    if len(resultantsInTTuplesList) == 0:

        return set()

    #print resultantsInITuplesList

    # Compute the roots.

    Zi = ZZ[str(iVariable)]

    Zt = ZZ[str(tVariable)]

    polynomialRootsSet = set()

    # First, solve in the second variable since resultants are in the first

    # variable.

    for resultantInTTuple in resultantsInTTuplesList:

        iRootsList = Zi(resultantInTTuple[2]).roots()

        # For each iRoot, compute the corresponding tRoots and check 

        # them in the input polynomial.

        for iRoot in iRootsList:

            #print "iRoot:", iRoot

            # Roots returned by root() are (value, multiplicity) tuples.

            tRootsList = \

                Zt(resultantInTTuple[0].subs({resultantInTTuple[0].variables()[0]:iRoot[0]})).roots()

            print tRootsList

            # The tRootsList can be empty, hence the test.

            if len(tRootsList) != 0:

                for tRoot in tRootsList:

                    polyEvalModN = inputPolynomial(iRoot[0],tRoot[0]) / N

                    # polyEvalModN must be an integer.

                    if polyEvalModN == int(polyEvalModN):

                        polynomialRootsSet.add((iRoot[0],tRoot[0]))

    return polynomialRootsSet

# End slz_compute_integer_polynomial_modular_roots_2.

#

def slz_compute_polynomial_and_interval(functionSo, degreeSo, lowerBoundSa, 

                                        upperBoundSa, approxAccurSa, 

                                        precSa=None):

    """

    Under the assumptions listed for slz_get_intervals_and_polynomials, compute

    a polynomial that approximates the function on a an interval starting

    at lowerBoundSa and finishing at a value that guarantees that the polynomial

    approximates with the expected precision.

    The interval upper bound is lowered until the expected approximation 

    precision is reached.

    The polynomial, the bounds, the center of the interval and the error

    are returned.

    OUTPUT:

    A tuple made of 4 Sollya objects:

    - a polynomial;

    - an range (an interval, not in the sense of number given as an interval);

    - the center of the interval;

    - the maximum error in the approximation of the input functionSo by the

      output polynomial ; this error <= approxAccurSaS.

    """

    #print"In slz_compute_polynomial_and_interval..."

    ## Superficial argument check.

    if lowerBoundSa > upperBoundSa:

        return None

    ## Change Sollya precision, if requested.

    if precSa is None:

        precSa = ceil((RR('1.5') * abs(RR(approxAccurSa).log2())) / 64) * 64

        #print "Computed internal precision:", precSa

        if precSa < 192:

            precSa = 192

    sollyaPrecChanged = False

    (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa()

    if precSa > initialSollyaPrecSa:

        if precSa <= 2:

            print inspect.stack()[0][3], ": precision change <=2 requested."

        pobyso_set_prec_sa_so(precSa)

        sollyaPrecChanged = True

    RRR = lowerBoundSa.parent()

    intervalShrinkConstFactorSa = RRR('0.9')

    absoluteErrorTypeSo = pobyso_absolute_so_so()

    currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa)

    currentUpperBoundSa = upperBoundSa

    currentLowerBoundSa = lowerBoundSa

    # What we want here is the polynomial without the variable change, 

    # since our actual variable will be x-intervalCenter defined over the 

    # domain [lowerBound-intervalCenter , upperBound-intervalCenter]. 

    (polySo, intervalCenterSo, maxErrorSo) = \

        pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, 

                                                    currentRangeSo, 

                                                    absoluteErrorTypeSo)

    maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    while maxErrorSa > approxAccurSa:

        print "++Approximation error:", maxErrorSa.n()

        sollya_lib_clear_obj(polySo)

        sollya_lib_clear_obj(intervalCenterSo)

        sollya_lib_clear_obj(maxErrorSo)

        # Very empirical shrinking factor.

        shrinkFactorSa = 1 /  (maxErrorSa/approxAccurSa).log2().abs()

        print "Shrink factor:", \

              shrinkFactorSa.n(), \

              intervalShrinkConstFactorSa

        print

        #errorRatioSa = approxAccurSa/maxErrorSa

        #print "Error ratio: ", errorRatioSa

        # Make sure interval shrinks.

        if shrinkFactorSa > intervalShrinkConstFactorSa:

            actualShrinkFactorSa = intervalShrinkConstFactorSa

            #print "Fixed"

        else:

            actualShrinkFactorSa = shrinkFactorSa

            #print "Computed",shrinkFactorSa,maxErrorSa

            #print shrinkFactorSa, maxErrorSa

        #print "Shrink factor", actualShrinkFactorSa

        currentUpperBoundSa = currentLowerBoundSa + \

                                (currentUpperBoundSa - currentLowerBoundSa) * \

                                actualShrinkFactorSa

        #print "Current upper bound:", currentUpperBoundSa

        sollya_lib_clear_obj(currentRangeSo)

        # Check what is left with the bounds.

        if currentUpperBoundSa <= currentLowerBoundSa or \

              currentUpperBoundSa == currentLowerBoundSa.nextabove():

            sollya_lib_clear_obj(absoluteErrorTypeSo)

            print "Can't find an interval."

            print "Use either or both a higher polynomial degree or a higher",

            print "internal precision."

            print "Aborting!"

            if sollyaPrecChanged:

                pobyso_set_prec_so_so(initialSollyaPrecSo)

            sollya_lib_clear_obj(initialSollyaPrecSo)

            return None

        currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa, 

                                                      currentUpperBoundSa)

        # print "New interval:",

        # pobyso_autoprint(currentRangeSo)

        #print "Second Taylor expansion call."

        (polySo, intervalCenterSo, maxErrorSo) = \

            pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, 

                                                        currentRangeSo, 

                                                        absoluteErrorTypeSo)

        #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR)

        #print "Max errorSo:",

        #pobyso_autoprint(maxErrorSo)

        maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

        #print "Max errorSa:", maxErrorSa

        #print "Sollya prec:",

        #pobyso_autoprint(sollya_lib_get_prec(None))

    # End while

    sollya_lib_clear_obj(absoluteErrorTypeSo)

    if sollyaPrecChanged:

        pobyso_set_prec_so_so(initialSollyaPrecSo)

    sollya_lib_clear_obj(initialSollyaPrecSo)

    return (polySo, currentRangeSo, intervalCenterSo, maxErrorSo)

# End slz_compute_polynomial_and_interval

def slz_compute_polynomial_and_interval_01(functionSo, degreeSo, lowerBoundSa, 

                                        upperBoundSa, approxAccurSa, 

                                        sollyaPrecSa=None, debug=False):

    """

    Under the assumptions listed for slz_get_intervals_and_polynomials, compute

    a polynomial that approximates the function on a an interval starting

    at lowerBoundSa and finishing at a value that guarantees that the polynomial

    approximates with the expected precision.

    The interval upper bound is lowered until the expected approximation 

    precision is reached.

    The polynomial, the bounds, the center of the interval and the error

    are returned.

    OUTPUT:

    A tuple made of 4 Sollya objects:

    - a polynomial;

    - an range (an interval, not in the sense of number given as an interval);

    - the center of the interval;

    - the maximum error in the approximation of the input functionSo by the

      output polynomial ; this error <= approxAccurSaS.

    """

    #print"In slz_compute_polynomial_and_interval..."

    ## Superficial argument check.

    if lowerBoundSa > upperBoundSa:

        print inspect.stack()[0][3], ": lower bound is larger than upper bound. "

        return None

    ## Change Sollya precision, if requested.

    (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa()

    sollyaPrecChangedSa = False

    if sollyaPrecSa is None:

        sollyaPrecSa = initialSollyaPrecSa

    else:

        if sollyaPrecSa > initialSollyaPrecSa:

            if sollyaPrecSa <= 2:

                print inspect.stack()[0][3], ": precision change <= 2 requested."

            pobyso_set_prec_sa_so(sollyaPrecSa)

            sollyaPrecChangedSa = True

    ## Other initializations and data recovery.

    RRR = lowerBoundSa.parent()

    intervalShrinkConstFactorSa = RRR('0.9')

    absoluteErrorTypeSo = pobyso_absolute_so_so()

    currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa)

    currentUpperBoundSa = upperBoundSa

    currentLowerBoundSa = lowerBoundSa

    # What we want here is the polynomial without the variable change, 

    # since our actual variable will be x-intervalCenter defined over the 

    # domain [lowerBound-intervalCenter , upperBound-intervalCenter]. 

    (polySo, intervalCenterSo, maxErrorSo) = \

        pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, 

                                                    currentRangeSo, 

                                                    absoluteErrorTypeSo)

    maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    while maxErrorSa > approxAccurSa:

        print "++Approximation error:", maxErrorSa.n()

        sollya_lib_clear_obj(polySo)

        sollya_lib_clear_obj(intervalCenterSo)

        sollya_lib_clear_obj(maxErrorSo)

        # Very empirical shrinking factor.

        shrinkFactorSa = 1 /  (maxErrorSa/approxAccurSa).log2().abs()

        print "Shrink factor:", \

              shrinkFactorSa.n(), \

              intervalShrinkConstFactorSa

        print

        #errorRatioSa = approxAccurSa/maxErrorSa

        #print "Error ratio: ", errorRatioSa

        # Make sure interval shrinks.

        if shrinkFactorSa > intervalShrinkConstFactorSa:

            actualShrinkFactorSa = intervalShrinkConstFactorSa

            #print "Fixed"

        else:

            actualShrinkFactorSa = shrinkFactorSa

            #print "Computed",shrinkFactorSa,maxErrorSa

            #print shrinkFactorSa, maxErrorSa

        #print "Shrink factor", actualShrinkFactorSa

        currentUpperBoundSa = currentLowerBoundSa + \

                                (currentUpperBoundSa - currentLowerBoundSa) * \

                                actualShrinkFactorSa

        #print "Current upper bound:", currentUpperBoundSa

        sollya_lib_clear_obj(currentRangeSo)

        # Check what is left with the bounds.

        if currentUpperBoundSa <= currentLowerBoundSa or \

              currentUpperBoundSa == currentLowerBoundSa.nextabove():

            sollya_lib_clear_obj(absoluteErrorTypeSo)

            print "Can't find an interval."

            print "Use either or both a higher polynomial degree or a higher",

            print "internal precision."

            print "Aborting!"

            if sollyaPrecChangedSa:

                pobyso_set_prec_so_so(initialSollyaPrecSo)

            sollya_lib_clear_obj(initialSollyaPrecSo)

            return None

        currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa, 

                                                      currentUpperBoundSa)

        # print "New interval:",

        # pobyso_autoprint(currentRangeSo)

        #print "Second Taylor expansion call."

        (polySo, intervalCenterSo, maxErrorSo) = \

            pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, 

                                                        currentRangeSo, 

                                                        absoluteErrorTypeSo)

        #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR)

        #print "Max errorSo:",

        #pobyso_autoprint(maxErrorSo)

        maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

        #print "Max errorSa:", maxErrorSa

        #print "Sollya prec:",

        #pobyso_autoprint(sollya_lib_get_prec(None))

    # End while

    sollya_lib_clear_obj(absoluteErrorTypeSo)

    itpSo           = pobyso_constant_from_int_sa_so(floor(sollyaPrecSa/3))

    ftpSo           = pobyso_constant_from_int_sa_so(floor(2*sollyaPrecSa/3))

    maxPrecSo       = pobyso_constant_from_int_sa_so(sollyaPrecSa)

    approxAccurSo   = pobyso_constant_sa_so(RR(approxAccurSa))

    if debug:

        print inspect.stack()[0][3], "SollyaPrecSa:", sollyaPrecSa

        print "About to call polynomial rounding with:"

        print "polySo: ",           ; pobyso_autoprint(polySo)

        print "functionSo: ",       ; pobyso_autoprint(functionSo)

        print "intervalCenterSo: ", ; pobyso_autoprint(intervalCenterSo)

        print "currentRangeSo: ",   ; pobyso_autoprint(currentRangeSo)

        print "itpSo: ",            ; pobyso_autoprint(itpSo)

        print "ftpSo: ",            ; pobyso_autoprint(ftpSo)

        print "maxPrecSo: ",        ; pobyso_autoprint(maxPrecSo)

        print "approxAccurSo: ",    ; pobyso_autoprint(approxAccurSo)

    """

    # Naive rounding.

    (roundedPolySo, roundedPolyMaxErrSo) = \

    pobyso_polynomial_coefficients_progressive_round_so_so(polySo,

                                                           functionSo,

                                                           intervalCenterSo,

                                                           currentRangeSo,

                                                           itpSo,

                                                           ftpSo,

                                                           maxPrecSo,

                                                           approxAccurSo)

    """

    # Proved rounding.

    (roundedPolySo, roundedPolyMaxErrSo) = \

        pobyso_round_coefficients_progressive_so_so(polySo, 

                                                    functionSo,

                                                    maxPrecSo,

                                                    currentRangeSo,

                                                    intervalCenterSo,

                                                    maxErrorSo,

                                                    approxAccurSo,

                                                    debug=False)

    #### Comment out the two next lines when polynomial rounding is activated.

    #roundedPolySo = sollya_lib_copy_obj(polySo)

    #roundedPolyMaxErrSo =  sollya_lib_copy_obj(maxErrorSo)

    sollya_lib_clear_obj(polySo)

    sollya_lib_clear_obj(maxErrorSo)

    sollya_lib_clear_obj(itpSo)

    sollya_lib_clear_obj(ftpSo)

    sollya_lib_clear_obj(approxAccurSo)

    if sollyaPrecChangedSa:

        pobyso_set_prec_so_so(initialSollyaPrecSo)

    sollya_lib_clear_obj(initialSollyaPrecSo)

    if debug:

        print "1: ", ; pobyso_autoprint(roundedPolySo)

        print "2: ", ; pobyso_autoprint(currentRangeSo)

        print "3: ", ; pobyso_autoprint(intervalCenterSo)

        print "4: ", ; pobyso_autoprint(roundedPolyMaxErrSo)

    return (roundedPolySo, currentRangeSo, intervalCenterSo, roundedPolyMaxErrSo)

# End slz_compute_polynomial_and_interval_01

def slz_compute_polynomial_and_interval_02(functionSo, degreeSo, lowerBoundSa, 

                                        upperBoundSa, approxAccurSa, 

                                        sollyaPrecSa=None, debug=True ):

    """

    Under the assumptions listed for slz_get_intervals_and_polynomials, compute

    a polynomial that approximates the function on a an interval starting

    at lowerBoundSa and finishing at a value that guarantees that the polynomial

    approximates with the expected precision.

    The interval upper bound is lowered until the expected approximation 

    precision is reached.

    The polynomial, the bounds, the center of the interval and the error

    are returned.

    OUTPUT:

    A tuple made of 4 Sollya objects:

    - a polynomial;

    - an range (an interval, not in the sense of number given as an interval);

    - the center of the interval;

    - the maximum error in the approximation of the input functionSo by the

      output polynomial ; this error <= approxAccurSaS.

    Changes fom v 01:

        extra verbose.

    """

    print"In slz_compute_polynomial_and_interval..."

    ## Superficial argument check.

    if lowerBoundSa > upperBoundSa:

        return None

    ## Change Sollya precision, if requested.

    sollyaPrecChanged = False

    (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa()

    #print "Initial Sollya prec:", initialSollyaPrecSa, type(initialSollyaPrecSa)

    if sollyaPrecSa is None:

        sollyaPrecSa = initialSollyaPrecSa

    else:

        if sollyaPrecSa <= 2:

            print inspect.stack()[0][3], ": precision change <=2 requested."

        pobyso_set_prec_sa_so(sollyaPrecSa)

        sollyaPrecChanged = True

    RRR = lowerBoundSa.parent()

    intervalShrinkConstFactorSa = RRR('0.9')

    absoluteErrorTypeSo = pobyso_absolute_so_so()

    currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa)

    currentUpperBoundSa = upperBoundSa

    currentLowerBoundSa = lowerBoundSa

    #pobyso_autoprint(functionSo)

    #pobyso_autoprint(degreeSo)

    #pobyso_autoprint(currentRangeSo)

    #pobyso_autoprint(absoluteErrorTypeSo)

    ## What we want here is the polynomial without the variable change, 

    #  since our actual variable will be x-intervalCenter defined over the 

    #  domain [lowerBound-intervalCenter , upperBound-intervalCenter].

    (polySo, intervalCenterSo, maxErrorSo) = \

        pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, 

                                                    currentRangeSo, 

                                                    absoluteErrorTypeSo)

    maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    print "...after Taylor expansion."

    while maxErrorSa > approxAccurSa:

        print "++Approximation error:", maxErrorSa.n()

        sollya_lib_clear_obj(polySo)

        sollya_lib_clear_obj(intervalCenterSo)

        sollya_lib_clear_obj(maxErrorSo)

        # Very empirical shrinking factor.

        shrinkFactorSa = 1 /  (maxErrorSa/approxAccurSa).log2().abs()

        print "Shrink factor:", \

              shrinkFactorSa.n(), \

              intervalShrinkConstFactorSa

        print

        #errorRatioSa = approxAccurSa/maxErrorSa

        #print "Error ratio: ", errorRatioSa

        # Make sure interval shrinks.

        if shrinkFactorSa > intervalShrinkConstFactorSa:

            actualShrinkFactorSa = intervalShrinkConstFactorSa

            #print "Fixed"

        else:

            actualShrinkFactorSa = shrinkFactorSa

            #print "Computed",shrinkFactorSa,maxErrorSa

            #print shrinkFactorSa, maxErrorSa

        #print "Shrink factor", actualShrinkFactorSa

        currentUpperBoundSa = currentLowerBoundSa + \

                                (currentUpperBoundSa - currentLowerBoundSa) * \

                                actualShrinkFactorSa

        #print "Current upper bound:", currentUpperBoundSa

        sollya_lib_clear_obj(currentRangeSo)

        # Check what is left with the bounds.

        if currentUpperBoundSa <= currentLowerBoundSa or \

              currentUpperBoundSa == currentLowerBoundSa.nextabove():

            sollya_lib_clear_obj(absoluteErrorTypeSo)

            print "Can't find an interval."

            print "Use either or both a higher polynomial degree or a higher",

            print "internal precision."

            print "Aborting!"

            if sollyaPrecChanged:

                pobyso_set_prec_so_so(initialSollyaPrecSo)

            sollya_lib_clear_obj(initialSollyaPrecSo)

            return None

        currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa, 

                                                      currentUpperBoundSa)

        # print "New interval:",

        # pobyso_autoprint(currentRangeSo)

        #print "Second Taylor expansion call."

        (polySo, intervalCenterSo, maxErrorSo) = \

            pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, 

                                                        currentRangeSo, 

                                                        absoluteErrorTypeSo)

        #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR)

        #print "Max errorSo:",

        #pobyso_autoprint(maxErrorSo)

        maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

        #print "Max errorSa:", maxErrorSa

        #print "Sollya prec:",

        #pobyso_autoprint(sollya_lib_get_prec(None))

    # End while

    sollya_lib_clear_obj(absoluteErrorTypeSo)

    itpSo           = pobyso_constant_from_int_sa_so(floor(sollyaPrecSa/3))

    ftpSo           = pobyso_constant_from_int_sa_so(floor(2*sollyaPrecSa/3))

    maxPrecSo       = pobyso_constant_from_int_sa_so(sollyaPrecSa)

    approxAccurSo   = pobyso_constant_sa_so(RR(approxAccurSa))

    print "About to call polynomial rounding with:"

    print "polySo: ",           ; pobyso_autoprint(polySo)

    print "functionSo: ",       ; pobyso_autoprint(functionSo)

    print "intervalCenterSo: ", ; pobyso_autoprint(intervalCenterSo)

    print "currentRangeSo: ",   ; pobyso_autoprint(currentRangeSo)

    print "itpSo: ",            ; pobyso_autoprint(itpSo)

    print "ftpSo: ",            ; pobyso_autoprint(ftpSo)

    print "maxPrecSo: ",        ; pobyso_autoprint(maxPrecSo)

    print "approxAccurSo: ",    ; pobyso_autoprint(approxAccurSo)

    (roundedPolySo, roundedPolyMaxErrSo) = \

    pobyso_round_coefficients_progressive_so_so(polySo,

                                                functionSo,

                                                maxPrecSo,

                                                currentRangeSo,

                                                intervalCenterSo,

                                                maxErrorSo,

                                                approxAccurSo,

                                                debug = True)

    sollya_lib_clear_obj(polySo)

    sollya_lib_clear_obj(maxErrorSo)

    sollya_lib_clear_obj(itpSo)

    sollya_lib_clear_obj(ftpSo)

    sollya_lib_clear_obj(approxAccurSo)

    if sollyaPrecChanged:

        pobyso_set_prec_so_so(initialSollyaPrecSo)

    sollya_lib_clear_obj(initialSollyaPrecSo)

    print "1: ", ; pobyso_autoprint(roundedPolySo)

    print "2: ", ; pobyso_autoprint(currentRangeSo)

    print "3: ", ; pobyso_autoprint(intervalCenterSo)

    print "4: ", ; pobyso_autoprint(roundedPolyMaxErrSo)

    return (roundedPolySo, currentRangeSo, intervalCenterSo, roundedPolyMaxErrSo)

# End slz_compute_polynomial_and_interval_02

def slz_compute_reduced_polynomial(matrixRow,

                                    knownMonomials,

                                    var1,

                                    var1Bound,

                                    var2,

                                    var2Bound):

    """

    Compute a polynomial from a single reduced matrix row.

    This function was introduced in order to avoid the computation of the

    all the polynomials  from the full matrix (even those built from rows 

    that do no verify the Coppersmith condition) as this may involves 

    expensive operations over (large) integers.

    """

    ## Check arguments.

    if len(knownMonomials) == 0:

        return None

    # varNounds can be zero since 0^0 returns 1.

    if (var1Bound < 0) or (var2Bound < 0):

        return None

    ## Initialisations. 

    polynomialRing = knownMonomials[0].parent() 

    currentPolynomial = polynomialRing(0)

    # TODO: use zip instead of indices.

    for colIndex in xrange(0, len(knownMonomials)):

        currentCoefficient = matrixRow[colIndex]

        if currentCoefficient != 0:

            #print "Current coefficient:", currentCoefficient

            currentMonomial = knownMonomials[colIndex]

            #print "Monomial as multivariate polynomial:", \

            #currentMonomial, type(currentMonomial)

            degreeInVar1 = currentMonomial.degree(var1)

            #print "Degree in var1", var1, ":", degreeInVar1

            degreeInVar2 = currentMonomial.degree(var2)

            #print "Degree in var2", var2, ":", degreeInVar2

            if degreeInVar1 > 0:

                currentCoefficient = \

                    currentCoefficient / (var1Bound^degreeInVar1)

                #print "varBound1 in degree:", var1Bound^degreeInVar1

                #print "Current coefficient(1)", currentCoefficient

            if degreeInVar2 > 0:

                currentCoefficient = \

                    currentCoefficient / (var2Bound^degreeInVar2)

                #print "Current coefficient(2)", currentCoefficient

            #print "Current reduced monomial:", (currentCoefficient * \

            #                                    currentMonomial)

            currentPolynomial += (currentCoefficient * currentMonomial)

            #print "Current polynomial:", currentPolynomial

        # End if

    # End for colIndex.

    #print "Type of the current polynomial:", type(currentPolynomial)

    return(currentPolynomial)

# End slz_compute_reduced_polynomial

#

def slz_compute_reduced_polynomials(reducedMatrix,

                                        knownMonomials,

                                        var1,

                                        var1Bound,

                                        var2,

                                        var2Bound):

    """

    Legacy function, use slz_compute_reduced_polynomials_list

    """

    return(slz_compute_reduced_polynomials_list(reducedMatrix,

                                                knownMonomials,

                                                var1,

                                                var1Bound,

                                                var2,

                                                var2Bound)

    )

#

def slz_compute_reduced_polynomials_list(reducedMatrix,

                                         knownMonomials,

                                         var1,

                                         var1Bound,

                                         var2,

                                         var2Bound):

    """

    From a reduced matrix, holding the coefficients, from a monomials list,

    from the bounds of each variable, compute the corresponding polynomials

    scaled back by dividing by the "right" powers of the variables bounds.

    The elements in knownMonomials must be of the "right" polynomial type.

    They set the polynomial type of the output polynomials in list.

    @param reducedMatrix:  the reduced matrix as output from LLL;

    @param kwnonMonomials: the ordered list of the monomials used to

                           build the polynomials;

    @param var1:           the first variable (of the "right" type);

    @param var1Bound:      the first variable bound;

    @param var2:           the second variable (of the "right" type);

    @param var2Bound:      the second variable bound.

    @return: a list of polynomials obtained with the reduced coefficients

             and scaled down with the bounds

    """

    # TODO: check input arguments.

    reducedPolynomials = []

    #print "type var1:", type(var1), " - type var2:", type(var2)

    for matrixRow in reducedMatrix.rows():

        currentPolynomial = 0

        for colIndex in xrange(0, len(knownMonomials)):

            currentCoefficient = matrixRow[colIndex]

            if currentCoefficient != 0:

                #print "Current coefficient:", currentCoefficient

                currentMonomial = knownMonomials[colIndex]

                parentRing = currentMonomial.parent()

                #print "Monomial as multivariate polynomial:", \

                #currentMonomial, type(currentMonomial)

                degreeInVar1 = currentMonomial.degree(parentRing(var1))

                #print "Degree in var", var1, ":", degreeInVar1

                degreeInVar2 = currentMonomial.degree(parentRing(var2))

                #print "Degree in var", var2, ":", degreeInVar2

                if degreeInVar1 > 0:

                    currentCoefficient /= var1Bound^degreeInVar1

                    #print "varBound1 in degree:", var1Bound^degreeInVar1

                    #print "Current coefficient(1)", currentCoefficient

                if degreeInVar2 > 0:

                    currentCoefficient /= var2Bound^degreeInVar2

                    #print "Current coefficient(2)", currentCoefficient

                #print "Current reduced monomial:", (currentCoefficient * \

                #                                    currentMonomial)

                currentPolynomial += (currentCoefficient * currentMonomial)

                #if degreeInVar1 == 0 and degreeInVar2 == 0:

                    #print "!!!! constant term !!!!"

                #print "Current polynomial:", currentPolynomial

            # End if

        # End for colIndex.

        #print "Type of the current polynomial:", type(currentPolynomial)

        reducedPolynomials.append(currentPolynomial)

    return reducedPolynomials 

# End slz_compute_reduced_polynomials_list.

def slz_compute_reduced_polynomials_list_from_rows(rowsList,

                                                   knownMonomials,

                                                   var1,

                                                   var1Bound,

                                                   var2,

                                                   var2Bound):

    """

    From a list of rows, holding the coefficients, from a monomials list,

    from the bounds of each variable, compute the corresponding polynomials

    scaled back by dividing by the "right" powers of the variables bounds.

    The elements in knownMonomials must be of the "right" polynomial type.

    They set the polynomial type of the output polynomials in list.

    @param rowsList:       the reduced matrix as output from LLL;

    @param kwnonMonomials: the ordered list of the monomials used to

                           build the polynomials;

    @param var1:           the first variable (of the "right" type);

    @param var1Bound:      the first variable bound;

    @param var2:           the second variable (of the "right" type);

    @param var2Bound:      the second variable bound.

    @return: a list of polynomials obtained with the reduced coefficients

             and scaled down with the bounds

    """

    # TODO: check input arguments.

    reducedPolynomials = []

    #print "type var1:", type(var1), " - type var2:", type(var2)

    for matrixRow in rowsList:

        currentPolynomial = 0

        for colIndex in xrange(0, len(knownMonomials)):

            currentCoefficient = matrixRow[colIndex]

            if currentCoefficient != 0:

                #print "Current coefficient:", currentCoefficient

                currentMonomial = knownMonomials[colIndex]

                parentRing = currentMonomial.parent()

                #print "Monomial as multivariate polynomial:", \

                #currentMonomial, type(currentMonomial)

                degreeInVar1 = currentMonomial.degree(parentRing(var1))

                #print "Degree in var", var1, ":", degreeInVar1

                degreeInVar2 = currentMonomial.degree(parentRing(var2))

                #print "Degree in var", var2, ":", degreeInVar2

                if degreeInVar1 > 0:

                    currentCoefficient /= var1Bound^degreeInVar1

                    #print "varBound1 in degree:", var1Bound^degreeInVar1

                    #print "Current coefficient(1)", currentCoefficient

                if degreeInVar2 > 0:

                    currentCoefficient /= var2Bound^degreeInVar2

                    #print "Current coefficient(2)", currentCoefficient

                #print "Current reduced monomial:", (currentCoefficient * \

                #                                    currentMonomial)

                currentPolynomial += (currentCoefficient * currentMonomial)

                #if degreeInVar1 == 0 and degreeInVar2 == 0:

                    #print "!!!! constant term !!!!"

                #print "Current polynomial:", currentPolynomial

            # End if

        # End for colIndex.

        #print "Type of the current polynomial:", type(currentPolynomial)

        reducedPolynomials.append(currentPolynomial)

    return reducedPolynomials 

# End slz_compute_reduced_polynomials_list_from_rows.

#

def slz_compute_scaled_function(functionSa,

                                lowerBoundSa,

                                upperBoundSa,

                                floatingPointPrecSa,

                                debug=False):

    """

    From a function, compute the scaled function whose domain

    is included in [1, 2) and whose image is also included in [1,2).

    Return a tuple: 

    [0]: the scaled function

    [1]: the scaled domain lower bound

    [2]: the scaled domain upper bound

    [3]: the scaled image lower bound

    [4]: the scaled image upper bound

    """

    ## The variable can be called anything.

    x = functionSa.variables()[0]

    # Scalling the domain -> [1,2[.

    boundsIntervalRifSa = RealIntervalField(floatingPointPrecSa)

    domainBoundsIntervalSa = boundsIntervalRifSa(lowerBoundSa, upperBoundSa)

    (invDomainScalingExpressionSa, domainScalingExpressionSa) = \

        slz_interval_scaling_expression(domainBoundsIntervalSa, x)

    if debug:

        print "domainScalingExpression for argument :", \

        invDomainScalingExpressionSa

        print "function: ", functionSa

    ff = functionSa.subs({x : domainScalingExpressionSa})

    ## Bless expression as a function.

    ff = ff.function(x)

    #ff = f.subs_expr(x==domainScalingExpressionSa)

    #domainScalingFunction(x) = invDomainScalingExpressionSa

    domainScalingFunction = invDomainScalingExpressionSa.function(x)

    scaledLowerBoundSa = \

        domainScalingFunction(lowerBoundSa).n(prec=floatingPointPrecSa)

    scaledUpperBoundSa = \

        domainScalingFunction(upperBoundSa).n(prec=floatingPointPrecSa)

    if debug:

        print 'ff:', ff, "- Domain:", scaledLowerBoundSa, \

              scaledUpperBoundSa

    #

    # Scalling the image -> [1,2[.

    flbSa = ff(scaledLowerBoundSa).n(prec=floatingPointPrecSa)

    fubSa = ff(scaledUpperBoundSa).n(prec=floatingPointPrecSa)

    if flbSa <= fubSa: # Increasing

        imageBinadeBottomSa = floor(flbSa.log2())

    else: # Decreasing

        imageBinadeBottomSa = floor(fubSa.log2())

    if debug:

        print 'ff:', ff, '- Image:', flbSa, fubSa, imageBinadeBottomSa

    imageBoundsIntervalSa = boundsIntervalRifSa(flbSa, fubSa)

    (invImageScalingExpressionSa,imageScalingExpressionSa) = \

        slz_interval_scaling_expression(imageBoundsIntervalSa, x)

    if debug:

        print "imageScalingExpression for argument :", \

              invImageScalingExpressionSa

    iis = invImageScalingExpressionSa.function(x)

    fff = iis.subs({x:ff})

    if debug:

        print "fff:", fff,

        print " - Image:", fff(scaledLowerBoundSa), fff(scaledUpperBoundSa)

    return([fff, scaledLowerBoundSa, scaledUpperBoundSa, \

            fff(scaledLowerBoundSa), fff(scaledUpperBoundSa)])

# End slz_compute_scaled_function

def slz_fix_bounds_for_binades(lowerBound, 

                               upperBound, 

                               func = None, 

                               domainDirection = -1,

                               imageDirection  = -1):

    """

    Assuming the function is increasing or decreasing over the 

    [lowerBound, upperBound] interval, return a lower bound lb and 

    an upper bound ub such that:

    - lb and ub belong to the same binade;

    - func(lb) and func(ub) belong to the same binade.

    domainDirection indicate how bounds move to fit into the same binade:

    - a negative value move the upper bound down;

    - a positive value move the lower bound up.

    imageDirection indicate how bounds move in order to have their image

    fit into the same binade, variation of the function is also condidered.

    For an increasing function:

    - negative value moves the upper bound down (and its image value as well);

    - a positive values moves the lower bound up (and its image value as well);

    For a decreasing function it is the other way round.

    """

    ## Arguments check

    if lowerBound > upperBound:

        return None

    if lowerBound == upperBound:

        return (lowerBound, upperBound)

    if func is None:

        return None

    #

    #varFunc = func.variables()[0]

    lb = lowerBound

    ub = upperBound

    lbBinade = slz_compute_binade(lb) 

    ubBinade = slz_compute_binade(ub)

    ## Domain binade.

    while lbBinade != ubBinade:

        newIntervalWidth = (ub - lb) / 2

        if domainDirection < 0:

            ub = lb + newIntervalWidth

            ubBinade = slz_compute_binade(ub)

        else:

            lb = lb + newIntervalWidth

            lbBinade = slz_compute_binade(lb) 

    ## Image binade.

    if lb == ub:

        return (lb, ub)

    lbImg = func(lb)

    ubImg = func(ub)

    funcIsInc = (ubImg >= lbImg)

    lbImgBinade = slz_compute_binade(lbImg)

    ubImgBinade = slz_compute_binade(ubImg)

    while lbImgBinade != ubImgBinade:

        newIntervalWidth = (ub - lb) / 2

        if imageDirection < 0:

            if funcIsInc:

                ub = lb + newIntervalWidth

                ubImgBinade = slz_compute_binade(func(ub))

                #print ubImgBinade

            else:

                lb = lb + newIntervalWidth

                lbImgBinade = slz_compute_binade(func(lb))

                #print lbImgBinade

        else:

            if funcIsInc:

                lb = lb + newIntervalWidth

                lbImgBinade = slz_compute_binade(func(lb))

                #print lbImgBinade

            else:

                ub = lb + newIntervalWidth

                ubImgBinade = slz_compute_binade(func(ub))

                #print ubImgBinade

    # End while lbImgBinade != ubImgBinade:

    return (lb, ub)

# End slz_fix_bounds_for_binades.

def slz_float_poly_of_float_to_rat_poly_of_rat(polyOfFloat):

    # Create a polynomial over the rationals.

    ratPolynomialRing = QQ[str(polyOfFloat.variables()[0])]

    return(ratPolynomialRing(polyOfFloat))

# End slz_float_poly_of_float_to_rat_poly_of_rat.

def slz_float_poly_of_float_to_rat_poly_of_rat_pow_two(polyOfFloat):

    """

     Create a polynomial over the rationals where all denominators are

     powers of two.

    """

    polyVariable = polyOfFloat.variables()[0] 

    RPR = QQ[str(polyVariable)]

    polyCoeffs      = polyOfFloat.coefficients()

    #print polyCoeffs

    polyExponents   = polyOfFloat.exponents()

    #print polyExponents

    polyDenomPtwoCoeffs = []

    for coeff in polyCoeffs:

        polyDenomPtwoCoeffs.append(sno_float_to_rat_pow_of_two_denom(coeff))

        #print "Converted coefficient:", sno_float_to_rat_pow_of_two_denom(coeff),

        #print type(sno_float_to_rat_pow_of_two_denom(coeff))

    ratPoly = RPR(0)

    #print type(ratPoly)

    ## !!! CAUTION !!! Do not use the RPR(coeff * polyVariagle^exponent)

    #  The coefficient becomes plainly wrong when exponent == 0.

    #  No clue as to why.

    for coeff, exponent in zip(polyDenomPtwoCoeffs, polyExponents):

        ratPoly += coeff * RPR(polyVariable^exponent)

    return ratPoly

# End slz_float_poly_of_float_to_rat_poly_of_rat.

def slz_get_intervals_and_polynomials(functionSa, degreeSa, 

                                      lowerBoundSa, 

                                      upperBoundSa, floatingPointPrecSa, 

                                      internalSollyaPrecSa, approxPrecSa):

    """

    Under the assumption that:

    - functionSa is monotonic on the [lowerBoundSa, upperBoundSa] interval;

    - lowerBound and upperBound belong to the same binade.

    from a:

    - function;

    - a degree

    - a pair of bounds;

    - the floating-point precision we work on;

    - the internal Sollya precision;

    - the requested approximation error

    The initial interval is, possibly, splitted into smaller intervals.

    It return a list of tuples, each made of:

    - a first polynomial (without the changed variable f(x) = p(x-x0));

    - a second polynomial (with a changed variable f(x) = q(x))

    - the approximation interval;

    - the center, x0, of the interval;

    - the corresponding approximation error.

    TODO: fix endless looping for some parameters sets.

    """

    resultArray = []

    # Set Sollya to the necessary internal precision.

    sollyaPrecChangedSa = False

    (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa()

    if internalSollyaPrecSa > currentSollyaPrecSa:

        if internalSollyaPrecSa <= 2:

            print inspect.stack()[0][3], ": precision change <=2 requested."

        pobyso_set_prec_sa_so(internalSollyaPrecSa)

        sollyaPrecChangedSa = True

    #

    x = functionSa.variables()[0] # Actual variable name can be anything.

    # Scaled function: [1=,2] -> [1,2].

    (fff, scaledLowerBoundSa, scaledUpperBoundSa,                             \

            scaledLowerBoundImageSa, scaledUpperBoundImageSa) =               \

                slz_compute_scaled_function(functionSa,                       \

                                            lowerBoundSa,                     \

                                            upperBoundSa,                     \

                                            floatingPointPrecSa)

    # In case bounds were in the negative real one may need to

    # switch scaled bounds.

    if scaledLowerBoundSa > scaledUpperBoundSa:

        scaledLowerBoundSa, scaledUpperBoundSa = \

            scaledUpperBoundSa, scaledLowerBoundSa

        #print "Switching!"

    print "Approximation accuracy: ", RR(approxAccurSa)

    # Prepare the arguments for the Taylor expansion computation with Sollya.

    functionSo = \

      pobyso_parse_string_sa_so(fff._assume_str().replace('_SAGE_VAR_', ''))

    degreeSo = pobyso_constant_from_int_sa_so(degreeSa)

    scaledBoundsSo = pobyso_bounds_to_range_sa_so(scaledLowerBoundSa, 

                                                  scaledUpperBoundSa)

    realIntervalField = RealIntervalField(max(lowerBoundSa.parent().precision(),

                                              upperBoundSa.parent().precision()))

    currentScaledLowerBoundSa = scaledLowerBoundSa

    currentScaledUpperBoundSa = scaledUpperBoundSa

    while True:

        ## Compute the first Taylor expansion.

        print "Computing a Taylor expansion..."

        (polySo, boundsSo, intervalCenterSo, maxErrorSo) = \

            slz_compute_polynomial_and_interval(functionSo, degreeSo,

                                        currentScaledLowerBoundSa, 

                                        currentScaledUpperBoundSa,

                                        approxAccurSa, internalSollyaPrecSa)

        print "...done."

        ## If slz_compute_polynomial_and_interval fails, it returns None.

        #  This value goes to the first variable: polySo. Other variables are

        #  not assigned and should not be tested.

        if polySo is None:

            print "slz_get_intervals_and_polynomials: Aborting and returning None!"

            if precChangedSa:

                pobyso_set_prec_so_so(initialSollyaPrecSo)

            sollya_lib_clear_obj(initialSollyaPrecSo)

            sollya_lib_clear_obj(functionSo)

            sollya_lib_clear_obj(degreeSo)

            sollya_lib_clear_obj(scaledBoundsSo)

            return None