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""" 

Educational Versions of Groebner Basis Algorithms. 

 

Following [BW93]_ the original Buchberger algorithm (c.f. algorithm 

GROEBNER in [BW93]_) and an improved version of Buchberger's algorithm 

(c.g. algorithm GROEBNERNEW2 in [BW93]_) are implemented. 

 

No attempt was made to optimize either algorithm as the emphasis of 

these implementations is a clean and easy presentation. To compute a 

Groebner basis in Sage efficiently use the 

:meth:`sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal.groebner_basis()` 

method on multivariate polynomial objects. 

 

 

.. note:: 

 

The notion of 'term' and 'monomial' in [BW93]_ is swapped from the 

notion of those words in Sage (or the other way around, however you 

prefer it). In Sage a term is a monomial multiplied by a 

coefficient, while in [BW93]_ a monomial is a term multiplied by a 

coefficient. Also, what is called LM (the leading monomial) in 

Sage is called HT (the head term) in [BW93]_. 

 

EXAMPLES: 

 

Consider Katsura-6 w.r.t. a ``degrevlex`` ordering.:: 

 

sage: from sage.rings.polynomial.toy_buchberger import * 

sage: P.<a,b,c,e,f,g,h,i,j,k> = PolynomialRing(GF(32003),10) 

sage: I = sage.rings.ideal.Katsura(P,6) 

 

sage: g1 = buchberger(I) 

sage: g2 = buchberger_improved(I) 

sage: g3 = I.groebner_basis() 

 

All algorithms actually compute a Groebner basis:: 

 

sage: Ideal(g1).basis_is_groebner() 

True 

sage: Ideal(g2).basis_is_groebner() 

True 

sage: Ideal(g3).basis_is_groebner() 

True 

 

The results are correct:: 

 

sage: Ideal(g1) == Ideal(g2) == Ideal(g3) 

True 

 

If ``get_verbose()`` is `>= 1` a protocol is provided:: 

 

sage: set_verbose(1) 

sage: P.<a,b,c> = PolynomialRing(GF(127),3) 

sage: I = sage.rings.ideal.Katsura(P) 

// sage... ideal 

 

sage: I 

Ideal (a + 2*b + 2*c - 1, a^2 + 2*b^2 + 2*c^2 - a, 2*a*b + 2*b*c - b) of Multivariate Polynomial Ring in a, b, c over Finite Field of size 127 

 

The original Buchberger algorithm performs 15 useless reductions to 

zero for this example:: 

 

sage: buchberger(I) 

(a + 2*b + 2*c - 1, a^2 + 2*b^2 + 2*c^2 - a) => -2*b^2 - 6*b*c - 6*c^2 + b + 2*c 

G: set([a + 2*b + 2*c - 1, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c]) 

<BLANKLINE> 

(a^2 + 2*b^2 + 2*c^2 - a, a + 2*b + 2*c - 1) => 0 

G: set([a + 2*b + 2*c - 1, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c]) 

<BLANKLINE> 

(a + 2*b + 2*c - 1, 2*a*b + 2*b*c - b) => -5*b*c - 6*c^2 - 63*b + 2*c 

G: set([a + 2*b + 2*c - 1, 2*a*b + 2*b*c - b, -5*b*c - 6*c^2 - 63*b + 2*c, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c]) 

<BLANKLINE> 

(2*a*b + 2*b*c - b, a + 2*b + 2*c - 1) => 0 

G: set([a + 2*b + 2*c - 1, 2*a*b + 2*b*c - b, -5*b*c - 6*c^2 - 63*b + 2*c, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c]) 

<BLANKLINE> 

(2*a*b + 2*b*c - b, -5*b*c - 6*c^2 - 63*b + 2*c) => -22*c^3 + 24*c^2 - 60*b - 62*c 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(2*a*b + 2*b*c - b, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a + 2*b + 2*c - 1, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a^2 + 2*b^2 + 2*c^2 - a, 2*a*b + 2*b*c - b) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(-2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a + 2*b + 2*c - 1, -5*b*c - 6*c^2 - 63*b + 2*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a^2 + 2*b^2 + 2*c^2 - a, -5*b*c - 6*c^2 - 63*b + 2*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(-5*b*c - 6*c^2 - 63*b + 2*c, -22*c^3 + 24*c^2 - 60*b - 62*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(-2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -22*c^3 + 24*c^2 - 60*b - 62*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(2*a*b + 2*b*c - b, -22*c^3 + 24*c^2 - 60*b - 62*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

(a^2 + 2*b^2 + 2*c^2 - a, -22*c^3 + 24*c^2 - 60*b - 62*c) => 0 

G: set([a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c]) 

<BLANKLINE> 

15 reductions to zero. 

[a + 2*b + 2*c - 1, -22*c^3 + 24*c^2 - 60*b - 62*c, 2*a*b + 2*b*c - b, a^2 + 2*b^2 + 2*c^2 - a, -2*b^2 - 6*b*c - 6*c^2 + b + 2*c, -5*b*c - 6*c^2 - 63*b + 2*c] 

 

The 'improved' Buchberger algorithm in contrast only performs 3 reductions to zero:: 

 

sage: buchberger_improved(I) 

(b^2 - 26*c^2 - 51*b + 51*c, b*c + 52*c^2 + 38*b + 25*c) => 11*c^3 - 12*c^2 + 30*b + 31*c 

G: set([a + 2*b + 2*c - 1, b^2 - 26*c^2 - 51*b + 51*c, 11*c^3 - 12*c^2 + 30*b + 31*c, b*c + 52*c^2 + 38*b + 25*c]) 

<BLANKLINE> 

(11*c^3 - 12*c^2 + 30*b + 31*c, b*c + 52*c^2 + 38*b + 25*c) => 0 

G: set([a + 2*b + 2*c - 1, b^2 - 26*c^2 - 51*b + 51*c, 11*c^3 - 12*c^2 + 30*b + 31*c, b*c + 52*c^2 + 38*b + 25*c]) 

<BLANKLINE> 

1 reductions to zero. 

[a + 2*b + 2*c - 1, b^2 - 26*c^2 - 51*b + 51*c, c^3 + 22*c^2 - 55*b + 49*c, b*c + 52*c^2 + 38*b + 25*c] 

 

REFERENCES: 

 

.. [BW93] Thomas Becker and Volker Weispfenning. *Groebner Bases - A 

Computational Approach To Commutative Algebra*. Springer, New York 

1993. 

 

AUTHOR: 

 

- Martin Albrecht (2007-05-24): initial version 

- Marshall Hampton (2009-07-08): some doctest additions 

""" 

from __future__ import print_function 

 

from sage.misc.misc import get_verbose 

from sage.arith.all import LCM 

from sage.structure.sequence import Sequence 

 

#some aliases that conform to Becker and Weispfenning's notation: 

LCM = lambda f,g: f.parent().monomial_lcm(f,g) 

LM = lambda f: f.lm() 

LT = lambda f: f.lt() 

 

def spol(f,g): 

""" 

Computes the S-polynomial of f and g. 

 

INPUT: 

 

- ``f,g`` - polynomials 

 

OUTPUT: 

 

- The S-polynomial of f and g. 

 

EXAMPLES:: 

 

sage: R.<x,y,z> = PolynomialRing(QQ,3) 

sage: from sage.rings.polynomial.toy_buchberger import spol 

sage: spol(x^2 - z - 1, z^2 - y - 1) 

x^2*y - z^3 + x^2 - z^2 

""" 

fg_lcm = LCM(LM(f),LM(g)) 

return fg_lcm//LT(f)*f - fg_lcm//LT(g)*g 

 

 

def buchberger(F): 

""" 

The original version of Buchberger's algorithm as presented in 

[BW93]_, page 214. 

 

INPUT: 

 

- ``F`` - an ideal in a multivariate polynomial ring 

 

OUTPUT: 

 

a Groebner basis for F 

 

.. note:: 

 

The verbosity of this function may be controlled with a 

``set_verbose()`` call. Any value >=1 will result in this 

function printing intermediate bases. 

 

EXAMPLES:: 

 

sage: from sage.rings.polynomial.toy_buchberger import buchberger 

sage: R.<x,y,z> = PolynomialRing(QQ,3) 

sage: set_verbose(0) 

sage: buchberger(R.ideal([x^2 - z - 1, z^2 - y - 1, x*y^2 - x - 1])) 

[-y^3 + x*z - x + y, y^2*z + y^2 - x - z - 1, x*y^2 - x - 1, x^2 - z - 1, z^2 - y - 1] 

 

""" 

G = set(F.gens()) 

B = set((g1, g2) for g1 in G for g2 in G if g1 != g2) 

 

if get_verbose() >= 1: 

reductions_to_zero = 0 

 

while B!=set(): 

g1,g2 = select(B) 

B.remove( (g1,g2) ) 

 

h = spol(g1,g2).reduce(G) 

if h != 0: 

B = B.union( [(g,h) for g in G] ) 

G.add( h ) 

 

if get_verbose() >= 1: 

print("(%s, %s) => %s" % (g1, g2, h)) 

print("G: %s\n" % G) 

if h == 0: 

reductions_to_zero += 1 

 

if get_verbose() >= 1: 

print("%d reductions to zero." % reductions_to_zero) 

 

return Sequence(G) 

 

def buchberger_improved(F): 

""" 

An improved version of Buchberger's algorithm as presented in 

[BW93]_, page 232. 

 

This variant uses the Gebauer-Moeller Installation to apply 

Buchberger's first and second criterion to avoid useless pairs. 

 

INPUT: 

 

- ``F`` - an ideal in a multivariate polynomial ring 

 

OUTPUT: 

 

a Groebner basis for F 

 

.. note:: 

 

The verbosity of this function may be controlled with a 

``set_verbose()`` call. Any value ``>=1`` will result in this 

function printing intermediate Groebner bases. 

 

EXAMPLES:: 

 

sage: from sage.rings.polynomial.toy_buchberger import buchberger_improved 

sage: R.<x,y,z> = PolynomialRing(QQ,3) 

sage: set_verbose(0) 

sage: buchberger_improved(R.ideal([x^4-y-z,x*y*z-1])) 

[x*y*z - 1, x^3 - y^2*z - y*z^2, y^3*z^2 + y^2*z^3 - x^2] 

""" 

F = inter_reduction(F.gens()) 

 

G = set() 

B = set() 

 

if get_verbose() >=1: 

reductions_to_zero = 0 

 

while F != set(): 

f = min(F) 

F.remove(f) 

G,B = update(G,B,f) 

 

while B != set(): 

 

g1,g2 = select(B) 

B.remove((g1,g2)) 

h = spol(g1,g2).reduce(G) 

if h!=0: G,B = update(G,B,h) 

 

if get_verbose() >= 1: 

print("(%s, %s) => %s" % (g1, g2, h)) 

print("G: %s\n" % G) 

if h == 0: 

reductions_to_zero += 1 

 

if get_verbose() >= 1: 

print("%d reductions to zero." % reductions_to_zero) 

 

return Sequence(inter_reduction(G)) 

 

def update(G,B,h): 

""" 

Update ``G`` using the list of critical pairs ``B`` and the 

polynomial ``h`` as presented in [BW93]_, page 230. For this, 

Buchberger's first and second criterion are tested. 

 

This function implements the Gebauer-Moeller Installation. 

 

INPUT: 

 

- ``G`` - an intermediate Groebner basis 

- ``B`` - a list of critical pairs 

- ``h`` - a polynomial 

 

OUTPUT: 

 

a tuple of an intermediate Groebner basis and a list of 

critical pairs 

 

EXAMPLES:: 

 

sage: from sage.rings.polynomial.toy_buchberger import update 

sage: R.<x,y,z> = PolynomialRing(QQ,3) 

sage: set_verbose(0) 

sage: update(set(),set(),x*y*z) 

({x*y*z}, set()) 

sage: G,B = update(set(),set(),x*y*z-1) 

sage: G,B = update(G,B,x*y^2-1) 

sage: G,B 

({x*y*z - 1, x*y^2 - 1}, {(x*y^2 - 1, x*y*z - 1)}) 

""" 

R = h.parent() 

 

C = set([(h,g) for g in G]) 

D = set() 

 

while C != set(): 

(h,g) = C.pop() 

 

lcm_divides = lambda rhs: R.monomial_divides( LCM(LM(h),LM(rhs[1])), LCM(LM(h),LM(g))) 

 

if R.monomial_pairwise_prime(LM(h),LM(g)) or \ 

(\ 

not any( lcm_divides(f) for f in C ) \ 

and 

not any( lcm_divides(f) for f in D ) \ 

): 

D.add( (h,g) ) 

 

E = set() 

 

while D != set(): 

(h,g) = D.pop() 

if not R.monomial_pairwise_prime(LM(h),LM(g)): 

E.add( (h,g) ) 

 

B_new = set() 

 

while B != set(): 

g1,g2 = B.pop() 

if not R.monomial_divides( LM(h), LCM(LM(g1),LM(g2)) ) or \ 

R.monomial_lcm(LM(g1),LM( h)) == LCM(LM(g1),LM(g2)) or \ 

R.monomial_lcm(LM( h),LM(g2)) == LCM(LM(g1),LM(g2)) : 

B_new.add( (g1,g2) ) 

 

B_new = B_new.union( E ) 

 

G_new = set() 

 

while G != set(): 

g = G.pop() 

if not R.monomial_divides(LM(h), LM(g)): 

G_new.add(g) 

 

G_new.add(h) 

 

return G_new,B_new 

 

def select(P): 

""" 

The normal selection strategy 

 

INPUT: 

 

- ``P`` - a list of critical pairs 

 

OUTPUT: 

 

an element of P 

 

EXAMPLES:: 

 

sage: from sage.rings.polynomial.toy_buchberger import select 

sage: R.<x,y,z> = PolynomialRing(QQ,3, order='lex') 

sage: ps = [x^3 - z -1, z^3 - y - 1, x^5 - y - 2] 

sage: pairs = [[ps[i],ps[j]] for i in range(3) for j in range(i+1,3)] 

sage: select(pairs) 

[x^3 - z - 1, -y + z^3 - 1] 

""" 

return min(P, key=lambda fi_fj: LCM(LM(fi_fj[0]), LM(fi_fj[1])).total_degree()) 

 

 

def inter_reduction(Q): 

""" 

If ``Q`` is the set `(f_1, ..., f_n)` this method 

returns `(g_1, ..., g_s)` such that: 

 

- `<f_1,...,f_n> = <g_1,...,g_s>` 

- `LM(g_i) != LM(g_j)` for all `i != j` 

- `LM(g_i)` does not divide `m` for all monomials `m` of 

`\{g_1,...,g_{i-1}, g_{i+1},...,g_s\}` 

- `LC(g_i) == 1` for all `i`. 

 

INPUT: 

 

- ``Q`` - a set of polynomials 

 

EXAMPLES:: 

 

sage: from sage.rings.polynomial.toy_buchberger import inter_reduction 

sage: inter_reduction(set()) 

set() 

 

:: 

 

sage: P.<x,y> = QQ[] 

sage: reduced = inter_reduction(set([x^2-5*y^2,x^3])) 

sage: reduced == set([x*y^2, x^2-5*y^2]) 

True 

sage: reduced == inter_reduction(set([2*(x^2-5*y^2),x^3])) 

True 

""" 

if not Q: 

return Q # if Q is empty we cannot get a base ring 

base_ring = next(iter(Q)).base_ring() 

 

Q = set(Q) 

while True: 

Qbar = set(Q) 

for p in sorted(Qbar): 

Q.remove(p) 

h = p.reduce(Q) 

if h!=0: 

Q.add(h) 

if Qbar == Q: 

if base_ring.is_field(): 

return set([f.lc()**(-1) * f for f in Qbar]) 

else: return Qbar