(f, ring)
return Gr
Computes Groebner basis for a set of polynomials in `K[X]`.

    Given a set of multivariate polynomials `F`, finds another
    set `G`, such that Ideal `F = Ideal G` and `G` is a reduced
    Groebner basis.

    The resulting basis is unique and has monic generators if the
    ground domains is a field. Otherwise the result is non-unique
    but Groebner bases over e.g. integers can be computed (if the
    input polynomials are monic).

    Groebner bases can be used to choose specific generators for a
    polynomial ideal. Because these bases are unique you can check
    for ideal equality by comparing the Groebner bases.  To see if
    one polynomial lies in an ideal, divide by the elements in the
    base and see if the remainder vanishes.

    They can also be used to solve systems of polynomial equations
    as,  by choosing lexicographic ordering,  you can eliminate one
    variable at a time, provided that the ideal is zero-dimensional
    (finite number of solutions).

    References
    ==========

    1. [Bose03]_
    2. [Giovini91]_
    3. [Ajwa95]_
    4. [Cox97]_

    Algorithm used: an improved version of Buchberger's algorithm
    as presented in T. Becker, V. Weispfenning, Groebner Bases: A
    Computational Approach to Commutative Algebra, Springer, 1993,
    page 232.
Computes Groebner basis for a set of polynomials in `K[X]`.
Computes Groebner basis for a set of polynomials in K [ X ] .
def _buchberger(f, ring):
    """
    Computes Groebner basis for a set of polynomials in `K[X]`.

    Given a set of multivariate polynomials `F`, finds another
    set `G`, such that Ideal `F = Ideal G` and `G` is a reduced
    Groebner basis.

    The resulting basis is unique and has monic generators if the
    ground domains is a field. Otherwise the result is non-unique
    but Groebner bases over e.g. integers can be computed (if the
    input polynomials are monic).

    Groebner bases can be used to choose specific generators for a
    polynomial ideal. Because these bases are unique you can check
    for ideal equality by comparing the Groebner bases.  To see if
    one polynomial lies in an ideal, divide by the elements in the
    base and see if the remainder vanishes.

    They can also be used to solve systems of polynomial equations
    as,  by choosing lexicographic ordering,  you can eliminate one
    variable at a time, provided that the ideal is zero-dimensional
    (finite number of solutions).

    References
    ==========

    1. [Bose03]_
    2. [Giovini91]_
    3. [Ajwa95]_
    4. [Cox97]_

    Algorithm used: an improved version of Buchberger's algorithm
    as presented in T. Becker, V. Weispfenning, Groebner Bases: A
    Computational Approach to Commutative Algebra, Springer, 1993,
    page 232.

    """
    order = ring.order
    domain = ring.domain

    monomial_mul = ring.monomial_mul
    monomial_div = ring.monomial_div
    monomial_lcm = ring.monomial_lcm

    def select(P):
        # normal selection strategy
        # select the pair with minimum LCM(LM(f), LM(g))
        pr = min(P, key=lambda pair: order(monomial_lcm(f[pair[0]].LM, f[pair[1]].LM)))
        return pr

    def normal(g, J):
        h = g.rem([ f[j] for j in J ])

        if not h:
            return None
        else:
            h = h.monic()

            if not h in I:
                I[h] = len(f)
                f.append(h)

            return h.LM, I[h]

    def update(G, B, ih):
        # update G using the set of critical pairs B and h
        # [BW] page 230
        h = f[ih]
        mh = h.LM

        # filter new pairs (h, g), g in G
        C = G.copy()
        D = set()

        while C:
            # select a pair (h, g) by popping an element from C
            ig = C.pop()
            g = f[ig]
            mg = g.LM
            LCMhg = monomial_lcm(mh, mg)

            def lcm_divides(ip):
                # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
                m = monomial_lcm(mh, f[ip].LM)
                return monomial_div(LCMhg, m)

            # HT(h) and HT(g) disjoint: mh*mg == LCMhg
            if monomial_mul(mh, mg) == LCMhg or (
                not any(lcm_divides(ipx) for ipx in C) and
                    not any(lcm_divides(pr[1]) for pr in D)):
                D.add((ih, ig))

        E = set()

        while D:
            # select h, g from D (h the same as above)
            ih, ig = D.pop()
            mg = f[ig].LM
            LCMhg = monomial_lcm(mh, mg)

            if not monomial_mul(mh, mg) == LCMhg:
                E.add((ih, ig))

        # filter old pairs
        B_new = set()

        while B:
            # select g1, g2 from B (-> CP)
            ig1, ig2 = B.pop()
            mg1 = f[ig1].LM
            mg2 = f[ig2].LM
            LCM12 = monomial_lcm(mg1, mg2)

            # if HT(h) does not divide lcm(HT(g1), HT(g2))
            if not monomial_div(LCM12, mh) or \
                monomial_lcm(mg1, mh) == LCM12 or \
                    monomial_lcm(mg2, mh) == LCM12:
                B_new.add((ig1, ig2))

        B_new |= E

        # filter polynomials
        G_new = set()

        while G:
            ig = G.pop()
            mg = f[ig].LM

            if not monomial_div(mg, mh):
                G_new.add(ig)

        G_new.add(ih)

        return G_new, B_new
        # end of update ################################

    if not f:
        return []

    # replace f with a reduced list of initial polynomials; see [BW] page 203
    f1 = f[:]

    while True:
        f = f1[:]
        f1 = []

        for i in range(len(f)):
            p = f[i]
            r = p.rem(f[:i])

            if r:
                f1.append(r.monic())

        if f == f1:
            break

    I = {}            # ip = I[p]; p = f[ip]
    F = set()         # set of indices of polynomials
    G = set()         # set of indices of intermediate would-be Groebner basis
    CP = set()        # set of pairs of indices of critical pairs

    for i, h in enumerate(f):
        I[h] = i
        F.add(i)

    #####################################
    # algorithm GROEBNERNEWS2 in [BW] page 232

    while F:
        # select p with minimum monomial according to the monomial ordering
        h = min([f[x] for x in F], key=lambda f: order(f.LM))
        ih = I[h]
        F.remove(ih)
        G, CP = update(G, CP, ih)

    # count the number of critical pairs which reduce to zero
    reductions_to_zero = 0

    while CP:
        ig1, ig2 = select(CP)
        CP.remove((ig1, ig2))

        h = spoly(f[ig1], f[ig2], ring)
        # ordering divisors is on average more efficient [Cox] page 111
        G1 = sorted(G, key=lambda g: order(f[g].LM))
        ht = normal(h, G1)

        if ht:
            G, CP = update(G, CP, ht[1])
        else:
            reductions_to_zero += 1

    ######################################
    # now G is a Groebner basis; reduce it
    Gr = set()

    for ig in G:
        ht = normal(f[ig], G - set([ig]))

        if ht:
            Gr.add(ht[1])

    Gr = [f[ig] for ig in Gr]

    # order according to the monomial ordering
    Gr = sorted(Gr, key=lambda f: order(f.LM), reverse=True)

    return Gr
def _buchberger ( f , ring ) : order = ring . order domain = ring . domain monomial_mul = ring . monomial_mul monomial_div = ring . monomial_div monomial_lcm = ring . monomial_lcm def select ( P ) : # normal selection strategy # select the pair with minimum LCM(LM(f), LM(g)) pr = min ( P , key = lambda pair : order ( monomial_lcm ( f [ pair [ 0 ] ] . LM , f [ pair [ 1 ] ] . LM ) ) ) return pr def normal ( g , J ) : h = g . rem ( [ f [ j ] for j in J ] ) if not h : return None else : h = h . monic ( ) if not h in I : I [ h ] = len ( f ) f . append ( h ) return h . LM , I [ h ] def update ( G , B , ih ) : # update G using the set of critical pairs B and h # [BW] page 230 h = f [ ih ] mh = h . LM # filter new pairs (h, g), g in G C = G . copy ( ) D = set ( ) while C : # select a pair (h, g) by popping an element from C ig = C . pop ( ) g = f [ ig ] mg = g . LM LCMhg = monomial_lcm ( mh , mg ) def lcm_divides ( ip ) : # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g)) m = monomial_lcm ( mh , f [ ip ] . LM ) return monomial_div ( LCMhg , m ) # HT(h) and HT(g) disjoint: mh*mg == LCMhg if monomial_mul ( mh , mg ) == LCMhg or ( not any ( lcm_divides ( ipx ) for ipx in C ) and not any ( lcm_divides ( pr [ 1 ] ) for pr in D ) ) : D . add ( ( ih , ig ) ) E = set ( ) while D : # select h, g from D (h the same as above) ih , ig = D . pop ( ) mg = f [ ig ] . LM LCMhg = monomial_lcm ( mh , mg ) if not monomial_mul ( mh , mg ) == LCMhg : E . add ( ( ih , ig ) ) # filter old pairs B_new = set ( ) while B : # select g1, g2 from B (-> CP) ig1 , ig2 = B . pop ( ) mg1 = f [ ig1 ] . LM mg2 = f [ ig2 ] . LM LCM12 = monomial_lcm ( mg1 , mg2 ) # if HT(h) does not divide lcm(HT(g1), HT(g2)) if not monomial_div ( LCM12 , mh ) or monomial_lcm ( mg1 , mh ) == LCM12 or monomial_lcm ( mg2 , mh ) == LCM12 : B_new . add ( ( ig1 , ig2 ) ) B_new |= E # filter polynomials G_new = set ( ) while G : ig = G . pop ( ) mg = f [ ig ] . LM if not monomial_div ( mg , mh ) : G_new . add ( ig ) G_new . add ( ih ) return G_new , B_new # end of update ################################ if not f : return [ ] # replace f with a reduced list of initial polynomials; see [BW] page 203 f1 = f [ : ] while True : f = f1 [ : ] f1 = [ ] for i in range ( len ( f ) ) : p = f [ i ] r = p . rem ( f [ : i ] ) if r : f1 . append ( r . monic ( ) ) if f == f1 : break I = { } # ip = I[p]; p = f[ip] F = set ( ) # set of indices of polynomials G = set ( ) # set of indices of intermediate would-be Groebner basis CP = set ( ) # set of pairs of indices of critical pairs for i , h in enumerate ( f ) : I [ h ] = i F . add ( i ) ##################################### # algorithm GROEBNERNEWS2 in [BW] page 232 while F : # select p with minimum monomial according to the monomial ordering h = min ( [ f [ x ] for x in F ] , key = lambda f : order ( f . LM ) ) ih = I [ h ] F . remove ( ih ) G , CP = update ( G , CP , ih ) # count the number of critical pairs which reduce to zero reductions_to_zero = 0 while CP : ig1 , ig2 = select ( CP ) CP . remove ( ( ig1 , ig2 ) ) h = spoly ( f [ ig1 ] , f [ ig2 ] , ring ) # ordering divisors is on average more efficient [Cox] page 111 G1 = sorted ( G , key = lambda g : order ( f [ g ] . LM ) ) ht = normal ( h , G1 ) if ht : G , CP = update ( G , CP , ht [ 1 ] ) else : reductions_to_zero += 1 ###################################### # now G is a Groebner basis; reduce it Gr = set ( ) for ig in G : ht = normal ( f [ ig ] , G - set ( [ ig ] ) ) if ht : Gr . add ( ht [ 1 ] ) Gr = [ f [ ig ] for ig in Gr ] # order according to the monomial ordering Gr = sorted ( Gr , key = lambda f : order ( f . LM ) , reverse = True ) return Gr