def select(P): # normal selection strategy # select the pair with minimum LCM(LM(f), LM(g)) pr = minkey(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u)))) return pr
def sdp_groebner(f, u, O, K, gens="", verbose=False): """ 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. Added optional ``gens`` argument to apply :func:`sdp_str` for the purpose of debugging the algorithm. """ if not K.has_Field: raise DomainError("can't compute a Groebner basis over %s" % K) def select(P): # normal selection strategy # select the pair with minimum LCM(LM(f), LM(g)) pr = minkey(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u)))) return pr def normal(g, J): h = sdp_rem(g, [f[j] for j in J], u, O, K) if not h: return None else: h = sdp_monic(h, K) h = tuple(h) if not h in I: I[h] = len(f) f.append(h) return sdp_LM(h, u), 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 = sdp_LM(h, u) # 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 = sdp_LM(g, u) LCMhg = monomial_lcm(mh, mg) def lcm_divides(ip): # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g)) m = monomial_lcm(mh, sdp_LM(f[ip], u)) 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 = sdp_LM(f[ig], u) 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 = sdp_LM(f[ig1], u) mg2 = sdp_LM(f[ig2], u) 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 = sdp_LM(f[ig], u) 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 = sdp_rem(p, f[:i], u, O, K) if r: f1.append(sdp_monic(r, K)) if f == f1: break f = [tuple(p) for p in f] 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 O h = minkey([f[x] for x in F], key=lambda f: O(sdp_LM(f, u))) 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 = sdp_spoly(f[ig1], f[ig2], u, O, K) # ordering divisors is on average more efficient [Cox] page 111 G1 = sorted(G, key=lambda g: O(sdp_LM(f[g], u))) 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 = [list(f[ig]) for ig in Gr] # order according to the monomial ordering Gr = sorted(Gr, key=lambda f: O(sdp_LM(f, u)), reverse=True) if verbose: print "reductions_to_zero = %d" % reductions_to_zero return Gr
def sdp_groebner(f, u, O, K, gens='', verbose=False): """ 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. Added optional ``gens`` argument to apply :func:`sdp_str` for the purpose of debugging the algorithm. """ if not K.has_Field: raise DomainError("can't compute a Groebner basis over %s" % K) def select(P): # select the pair with minimum LCM(LM(f), LM(g)) pr = minkey(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u)))) return pr def normal(g, J): h = sdp_rem(g, [f[j] for j in J], u, O, K) if not h: return None else: h = tuple(h) if not h in I: I[h] = len(f) f.append(h) return sdp_LM(h, u), 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 = sdp_LM(h, u) # 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 = sdp_LM(g, u) LCMhg = monomial_lcm(mh, mg) def lcm_divides(ip): # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g)) m = monomial_lcm(mh, sdp_LM(f[ip], u)) 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 = sdp_LM(f[ig], u) 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 = sdp_LM(f[ig1], u) mg2 = sdp_LM(f[ig2], u) 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 = sdp_LM(f[ig], u) 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 = sdp_rem(p, f[:i], u, O, K) if r: f1.append(r) if f == f1: break f = [tuple(p) for p in f] 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 O h = minkey([f[x] for x in F], key=lambda f: O(sdp_LM(f, u))) 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 = sdp_spoly(f[ig1], f[ig2], u, O, K) ht = normal(h, G) 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 = [sdp_monic(list(f[ig]), K) for ig in Gr] # order according to the monomial ordering Gr = sorted(Gr, key=lambda f: O(sdp_LM(f, u)), reverse=True) if verbose: print 'reductions_to_zero = %d' % reductions_to_zero return Gr
def select(P): # select the pair with minimum LCM(LM(f), LM(g)) pr = minkey(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u)))) return pr
def select(P): # normal selection strategy # select the pair with minimum LCM(LM(f), LM(g)) pr = minkey(P, key=lambda pair: O(monomial_lcm(sdp_LM(f[pair[0]], u), sdp_LM(f[pair[1]], u)))) return pr