예제 #1
0
def descendant_subgroups(S, C, R1_c_list, x, R2, N, Y):
    A_dict = C.A_dict
    A_dict_inv = C.A_dict_inv
    if C.is_complete():
        # if C is complete then it only needs to test
        # whether the relators in R2 are satisfied
        for w, alpha in product(R2, C.omega):
            if not C.scan_check(alpha, w):
                return
        # relators in R2 are satisfied, append the table to list
        S.append(C)
    else:
        # find the first undefined entry in Coset Table
        for alpha, x in product(range(len(C.table)), C.A):
            if C.table[alpha][A_dict[x]] is None:
                # this is "x" in pseudo-code (using "y" makes it clear)
                undefined_coset, undefined_gen = alpha, x
                break
        # for filling up the undefine entry we try all possible values
        # of β ∈ Ω or β = n where β^(undefined_gen^-1) is undefined
        reach = C.omega + [C.n]
        for beta in reach:
            if beta < N:
                if beta == C.n or C.table[beta][
                        A_dict_inv[undefined_gen]] is None:
                    try_descendant(S, C, R1_c_list, R2, N, undefined_coset, \
                            undefined_gen, beta, Y)
예제 #2
0
def descendant_subgroups(S, C, R1_c_list, x, R2, N, Y):
    A_dict = C.A_dict
    A_dict_inv = C.A_dict_inv
    if C.is_complete():
        # if C is complete then it only needs to test
        # whether the relators in R2 are satisfied
        for w, alpha in product(R2, C.omega):
            if not C.scan_check(alpha, w):
                return
        # relators in R2 are satisfied, append the table to list
        S.append(C)
    else:
        # find the first undefined entry in Coset Table
        for alpha, x in product(range(len(C.table)), C.A):
            if C.table[alpha][A_dict[x]] is None:
                # this is "x" in pseudo-code (using "y" makes it clear)
                undefined_coset, undefined_gen = alpha, x
                break
        # for filling up the undefine entry we try all possible values
        # of β ∈ Ω or β = n where β^(undefined_gen^-1) is undefined
        reach = C.omega + [C.n]
        for beta in reach:
            if beta < N:
                if beta == C.n or C.table[beta][A_dict_inv[undefined_gen]] is None:
                    try_descendant(S, C, R1_c_list, R2, N, undefined_coset, \
                            undefined_gen, beta, Y)
예제 #3
0
    def update(f, sugar, P):
        """Add f with sugar ``sugar`` to S, update P."""
        if not f:
            return P
        k = len(S)
        S.append(f)
        Sugars.append(sugar)

        LMf = sdm_LM(f)

        def removethis(pair):
            i, j, s, t = pair
            if LMf[0] != t[0]:
                return False
            tik = sdm_monomial_lcm(LMf, sdm_LM(S[i]))
            tjk = sdm_monomial_lcm(LMf, sdm_LM(S[j]))
            return tik != t and tjk != t and sdm_monomial_divides(tik, t) and \
                sdm_monomial_divides(tjk, t)

        # apply the chain criterion
        P = [p for p in P if not removethis(p)]

        # new-pair set
        N = [(i, k, Ssugar(i, k), sdm_monomial_lcm(LMf, sdm_LM(S[i])))
             for i in range(k) if LMf[0] == sdm_LM(S[i])[0]]
        # TODO apply the product criterion?
        N.sort(key=ourkey)
        remove = set()
        for i, p in enumerate(N):
            for j in range(i + 1, len(N)):
                if sdm_monomial_divides(p[3], N[j][3]):
                    remove.add(j)

        # TODO mergesort?
        P.extend(reversed([p for i, p in enumerate(N) if not i in remove]))
        P.sort(key=ourkey, reverse=True)
        # NOTE reverse-sort, because we want to pop from the end
        return P
예제 #4
0
    def update(f, sugar, P):
        """Add f with sugar ``sugar`` to S, update P."""
        if not f:
            return P
        k = len(S)
        S.append(f)
        Sugars.append(sugar)

        LMf = sdm_LM(f)

        def removethis(pair):
            i, j, s, t = pair
            if LMf[0] != t[0]:
                return False
            tik = sdm_monomial_lcm(LMf, sdm_LM(S[i]))
            tjk = sdm_monomial_lcm(LMf, sdm_LM(S[j]))
            return tik != t and tjk != t and sdm_monomial_divides(tik, t) and \
                sdm_monomial_divides(tjk, t)
        # apply the chain criterion
        P = [p for p in P if not removethis(p)]

        # new-pair set
        N = [(i, k, Ssugar(i, k), sdm_monomial_lcm(LMf, sdm_LM(S[i])))
             for i in range(k) if LMf[0] == sdm_LM(S[i])[0]]
        # TODO apply the product criterion?
        N.sort(key=ourkey)
        remove = set()
        for i, p in enumerate(N):
            for j in range(i + 1, len(N)):
                if sdm_monomial_divides(p[3], N[j][3]):
                    remove.add(j)

        # TODO mergesort?
        P.extend(reversed([p for i, p in enumerate(N) if not i in remove]))
        P.sort(key=ourkey, reverse=True)
        # NOTE reverse-sort, because we want to pop from the end
        return P
예제 #5
0
def sdm_groebner(G, NF, O, K):
    """
    Compute a minimal standard basis of ``G`` with respect to order ``O``.

    The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``.
    The ground field is assumed to be ``K``, and monomials ordered according
    to ``O``.

    Let `N` denote the submodule generated by elements of `G`. A standard
    basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for
    any subset `X` of `F`, `in(X)` denotes the submodule generated by the
    initial forms of elements of `X`. [SCA, defn 2.3.2]

    A standard basis is called minimal if no subset of it is a standard basis.

    One may show that standard bases are always generating sets.

    Minimal standard bases are not unique. This algorithm computes a
    deterministic result, depending on the particular order of `G`.

    See [SCA, algorithm 2.3.8, and remark 1.6.3].
    """
    # First compute a standard basis
    S = [f for f in G if f]
    P = list(combinations(S, 2))

    def prune(P, S, h):
        """
        Prune the pair-set by applying the chain criterion
        [SCA, remark 2.5.11].
        """
        remove = set()
        retain = set()
        for (a, b, c) in permutations(S, 3):
            A = sdm_LM(a)
            B = sdm_LM(b)
            C = sdm_LM(c)
            if len(set([A[0], B[0], C[0]])) != 1 or not h in [a, b, c] or \
               any(tuple(x) in retain for x in [a, b, c]):
                continue
            if monomial_divides(B[1:], monomial_lcm(A[1:], C[1:])):
                remove.add((tuple(a), tuple(c)))
                retain.update([tuple(b), tuple(c), tuple(a)])
        return [(f, g) for (f, g) in P if (h not in [f, g]) or \
                    ((tuple(f), tuple(g)) not in remove and \
                     (tuple(g), tuple(f)) not in remove)]

    while P:
        # TODO better data structures!!!
        #print len(P), len(S)
        # Use the "normal selection strategy"
        lcms = [(i, sdm_LM(f)[:1] + monomial_lcm(sdm_LM(f)[1:], sdm_LM(g)[1:])) for \
                i, (f, g) in enumerate(P)]
        i = min(lcms, key=lambda x: O(x[1]))[0]
        f, g = P.pop(i)
        h = NF(sdm_spoly(f, g, O, K), S, O, K)
        if h:
            S.append(h)
            P.extend((h, f) for f in S if sdm_LM(h)[0] == sdm_LM(f)[0])
            P = prune(P, S, h)

    # Now interreduce it. (TODO again, better data structures)
    S = set(tuple(f) for f in S)
    for a, b in permutations(S, 2):
        A = sdm_LM(list(a))
        B = sdm_LM(list(b))
        if sdm_monomial_divides(A, B) and b in S and a in S:
            S.remove(b)

    return sorted((list(f) for f in S), key=lambda f: O(sdm_LM(f)),
                  reverse=True)
예제 #6
0
def sdm_groebner(G, NF, O, K):
    """
    Compute a minimal standard basis of ``G`` with respect to order ``O``.

    The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``.
    The ground field is assumed to be ``K``, and monomials ordered according
    to ``O``.

    Let `N` denote the submodule generated by elements of `G`. A standard
    basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for
    any subset `X` of `F`, `in(X)` denotes the submodule generated by the
    initial forms of elements of `X`. [SCA, defn 2.3.2]

    A standard basis is called minimal if no subset of it is a standard basis.

    One may show that standard bases are always generating sets.

    Minimal standard bases are not unique. This algorithm computes a
    deterministic result, depending on the particular order of `G`.

    See [SCA, algorithm 2.3.8, and remark 1.6.3].
    """
    # First compute a standard basis
    S = [f for f in G if f]
    P = list(combinations(S, 2))

    def prune(P, S, h):
        """
        Prune the pair-set by applying the chain criterion
        [SCA, remark 2.5.11].
        """
        remove = set()
        retain = set()
        for (a, b, c) in permutations(S, 3):
            A = sdm_LM(a)
            B = sdm_LM(b)
            C = sdm_LM(c)
            if len(set([A[0], B[0], C[0]])) != 1 or not h in [a, b, c] or \
               any(tuple(x) in retain for x in [a, b, c]):
                continue
            if monomial_divides(B[1:], monomial_lcm(A[1:], C[1:])):
                remove.add((tuple(a), tuple(c)))
                retain.update([tuple(b), tuple(c), tuple(a)])
        return [(f, g) for (f, g) in P if (h not in [f, g]) or \
                    ((tuple(f), tuple(g)) not in remove and \
                     (tuple(g), tuple(f)) not in remove)]

    while P:
        # TODO better data structures!!!
        #print len(P), len(S)
        # Use the "normal selection strategy"
        lcms = [(i, sdm_LM(f)[:1] + monomial_lcm(sdm_LM(f)[1:], sdm_LM(g)[1:])) for \
                i, (f, g) in enumerate(P)]
        i = min(lcms, key=lambda x: O(x[1]))[0]
        f, g = P.pop(i)
        h = NF(sdm_spoly(f, g, O, K), S, O, K)
        if h:
            S.append(h)
            P.extend((h, f) for f in S if sdm_LM(h)[0] == sdm_LM(f)[0])
            P = prune(P, S, h)

    # Now interreduce it. (TODO again, better data structures)
    S = set(tuple(f) for f in S)
    for a, b in permutations(S, 2):
        A = sdm_LM(list(a))
        B = sdm_LM(list(b))
        if sdm_monomial_divides(A, B) and b in S and a in S:
            S.remove(b)

    return sorted((list(f) for f in S),
                  key=lambda f: O(sdm_LM(f)),
                  reverse=True)