Educational version of the \(d\)-Groebner basis algorithm over PIDs¶
No attempt was made to optimize this algorithm as the emphasis of this implementation is a clean and easy presentation.
Note
The notion of ‘term’ and ‘monomial’ in [BW1993] 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 [BW1993] 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 [BW1993].
EXAMPLES:
sage: from sage.rings.polynomial.toy_d_basis import d_basis
>>> from sage.all import *
>>> from sage.rings.polynomial.toy_d_basis import d_basis
from sage.rings.polynomial.toy_d_basis import d_basis
First, consider an example from arithmetic geometry:
sage: A.<x,y> = PolynomialRing(ZZ, 2)
sage: B.<X,Y> = PolynomialRing(Rationals(), 2)
sage: f = -y^2 - y + x^3 + 7*x + 1
sage: fx = f.derivative(x)
sage: fy = f.derivative(y)
sage: I = B.ideal([B(f), B(fx), B(fy)])
sage: I.groebner_basis() # needs sage.libs.singular
[1]
>>> from sage.all import *
>>> A = PolynomialRing(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> B = PolynomialRing(Rationals(), Integer(2), names=('X', 'Y',)); (X, Y,) = B._first_ngens(2)
>>> f = -y**Integer(2) - y + x**Integer(3) + Integer(7)*x + Integer(1)
>>> fx = f.derivative(x)
>>> fy = f.derivative(y)
>>> I = B.ideal([B(f), B(fx), B(fy)])
>>> I.groebner_basis() # needs sage.libs.singular
[1]
A.<x,y> = PolynomialRing(ZZ, 2) B.<X,Y> = PolynomialRing(Rationals(), 2) f = -y^2 - y + x^3 + 7*x + 1 fx = f.derivative(x) fy = f.derivative(y) I = B.ideal([B(f), B(fx), B(fy)]) I.groebner_basis() # needs sage.libs.singular
Since the output is 1, we know that there are no generic singularities.
To look at the singularities of the arithmetic surface, we need to do the corresponding computation over \(\ZZ\):
sage: I = A.ideal([f, fx, fy])
sage: gb = d_basis(I); gb # needs sage.libs.singular
[x - 2020, y - 11313, 22627]
sage: gb[-1].factor() # needs sage.libs.singular
11^3 * 17
>>> from sage.all import *
>>> I = A.ideal([f, fx, fy])
>>> gb = d_basis(I); gb # needs sage.libs.singular
[x - 2020, y - 11313, 22627]
>>> gb[-Integer(1)].factor() # needs sage.libs.singular
11^3 * 17
I = A.ideal([f, fx, fy]) gb = d_basis(I); gb # needs sage.libs.singular gb[-1].factor() # needs sage.libs.singular
This Groebner Basis gives a lot of information. First, the only fibers (over \(\ZZ\)) that are not smooth are at 11 = 0, and 17 = 0. Examining the Groebner Basis, we see that we have a simple node in both the fiber at 11 and at 17. From the factorization, we see that the node at 17 is regular on the surface (an \(I_1\) node), but the node at 11 is not. After blowing up this non-regular point, we find that it is an \(I_3\) node.
Another example. This one is from the Magma Handbook:
sage: # needs sage.libs.singular
sage: P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='lex')
sage: I = ideal(x^2 - 1, y^2 - 1, 2*x*y - z)
sage: I = Ideal(d_basis(I))
sage: x.reduce(I)
x
sage: (2*x).reduce(I)
y*z
>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P = PolynomialRing(IntegerRing(), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = ideal(x**Integer(2) - Integer(1), y**Integer(2) - Integer(1), Integer(2)*x*y - z)
>>> I = Ideal(d_basis(I))
>>> x.reduce(I)
x
>>> (Integer(2)*x).reduce(I)
y*z
# needs sage.libs.singular P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='lex') I = ideal(x^2 - 1, y^2 - 1, 2*x*y - z) I = Ideal(d_basis(I)) x.reduce(I) (2*x).reduce(I)
To compute modulo 4, we can add the generator 4 to our basis.:
sage: # needs sage.libs.singular
sage: I = ideal(x^2 - 1, y^2 - 1, 2*x*y - z, 4)
sage: gb = d_basis(I)
sage: R = P.change_ring(IntegerModRing(4))
sage: gb = [R(f) for f in gb if R(f)]; gb
[x^2 - 1, x*z + 2*y, 2*x - y*z, y^2 - 1, z^2, 2*z]
>>> from sage.all import *
>>> # needs sage.libs.singular
>>> I = ideal(x**Integer(2) - Integer(1), y**Integer(2) - Integer(1), Integer(2)*x*y - z, Integer(4))
>>> gb = d_basis(I)
>>> R = P.change_ring(IntegerModRing(Integer(4)))
>>> gb = [R(f) for f in gb if R(f)]; gb
[x^2 - 1, x*z + 2*y, 2*x - y*z, y^2 - 1, z^2, 2*z]
# needs sage.libs.singular I = ideal(x^2 - 1, y^2 - 1, 2*x*y - z, 4) gb = d_basis(I) R = P.change_ring(IntegerModRing(4)) gb = [R(f) for f in gb if R(f)]; gb
A third example is also from the Magma Handbook.
This example shows how one can use Groebner bases over the integers to find the primes modulo which a system of equations has a solution, when the system has no solutions over the rationals.
We first form a certain ideal \(I\) in \(\ZZ[x, y, z]\), and note that the Groebner basis of \(I\) over \(\QQ\) contains 1, so there are no solutions over \(\QQ\) or an algebraic closure of it (this is not surprising as there are 4 equations in 3 unknowns).
sage: P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='degneglex')
sage: I = ideal(x^2 - 3*y, y^3 - x*y, z^3 - x, x^4 - y*z + 1)
sage: I.change_ring(P.change_ring(RationalField())).groebner_basis() # needs sage.libs.singular
[1]
>>> from sage.all import *
>>> P = PolynomialRing(IntegerRing(), Integer(3), order='degneglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = ideal(x**Integer(2) - Integer(3)*y, y**Integer(3) - x*y, z**Integer(3) - x, x**Integer(4) - y*z + Integer(1))
>>> I.change_ring(P.change_ring(RationalField())).groebner_basis() # needs sage.libs.singular
[1]
P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='degneglex') I = ideal(x^2 - 3*y, y^3 - x*y, z^3 - x, x^4 - y*z + 1) I.change_ring(P.change_ring(RationalField())).groebner_basis() # needs sage.libs.singular
However, when we compute the Groebner basis of I (defined over \(\ZZ\)), we note that there is a certain integer in the ideal which is not 1:
sage: gb = d_basis(I); gb # needs sage.libs.singular
[z ..., y ..., x ..., 282687803443]
>>> from sage.all import *
>>> gb = d_basis(I); gb # needs sage.libs.singular
[z ..., y ..., x ..., 282687803443]
gb = d_basis(I); gb # needs sage.libs.singular
Now for each prime \(p\) dividing this integer 282687803443, the Groebner basis of I modulo \(p\) will be non-trivial and will thus give a solution of the original system modulo \(p\).:
sage: factor(282687803443)
101 * 103 * 27173681
sage: I.change_ring(P.change_ring(GF(101))).groebner_basis() # needs sage.libs.singular
[z - 33, y + 48, x + 19]
sage: I.change_ring(P.change_ring(GF(103))).groebner_basis() # needs sage.libs.singular
[z - 18, y + 8, x + 39]
sage: I.change_ring(P.change_ring(GF(27173681))).groebner_basis() # needs sage.libs.singular sage.rings.finite_rings
[z + 10380032, y + 3186055, x - 536027]
>>> from sage.all import *
>>> factor(Integer(282687803443))
101 * 103 * 27173681
>>> I.change_ring(P.change_ring(GF(Integer(101)))).groebner_basis() # needs sage.libs.singular
[z - 33, y + 48, x + 19]
>>> I.change_ring(P.change_ring(GF(Integer(103)))).groebner_basis() # needs sage.libs.singular
[z - 18, y + 8, x + 39]
>>> I.change_ring(P.change_ring(GF(Integer(27173681)))).groebner_basis() # needs sage.libs.singular sage.rings.finite_rings
[z + 10380032, y + 3186055, x - 536027]
factor(282687803443) I.change_ring(P.change_ring(GF(101))).groebner_basis() # needs sage.libs.singular I.change_ring(P.change_ring(GF(103))).groebner_basis() # needs sage.libs.singular I.change_ring(P.change_ring(GF(27173681))).groebner_basis() # needs sage.libs.singular sage.rings.finite_rings
Of course, modulo any other prime the Groebner basis is trivial so there are no other solutions. For example:
sage: I.change_ring(P.change_ring(GF(3))).groebner_basis() # needs sage.libs.singular
[1]
>>> from sage.all import *
>>> I.change_ring(P.change_ring(GF(Integer(3)))).groebner_basis() # needs sage.libs.singular
[1]
I.change_ring(P.change_ring(GF(3))).groebner_basis() # needs sage.libs.singular
AUTHOR:
Martin Albrecht (2008-08): initial version
- sage.rings.polynomial.toy_d_basis.d_basis(F, strat=True)[source]¶
Return the \(d\)-basis for the Ideal
Fas defined in [BW1993].INPUT:
F– an idealstrat– use update strategy (default:True)
EXAMPLES:
sage: from sage.rings.polynomial.toy_d_basis import d_basis sage: A.<x,y> = PolynomialRing(ZZ, 2) sage: f = -y^2 - y + x^3 + 7*x + 1 sage: fx = f.derivative(x) sage: fy = f.derivative(y) sage: I = A.ideal([f,fx,fy]) sage: gb = d_basis(I); gb # needs sage.libs.singular [x - 2020, y - 11313, 22627]
>>> from sage.all import * >>> from sage.rings.polynomial.toy_d_basis import d_basis >>> A = PolynomialRing(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> f = -y**Integer(2) - y + x**Integer(3) + Integer(7)*x + Integer(1) >>> fx = f.derivative(x) >>> fy = f.derivative(y) >>> I = A.ideal([f,fx,fy]) >>> gb = d_basis(I); gb # needs sage.libs.singular [x - 2020, y - 11313, 22627]
from sage.rings.polynomial.toy_d_basis import d_basis A.<x,y> = PolynomialRing(ZZ, 2) f = -y^2 - y + x^3 + 7*x + 1 fx = f.derivative(x) fy = f.derivative(y) I = A.ideal([f,fx,fy]) gb = d_basis(I); gb # needs sage.libs.singular
- sage.rings.polynomial.toy_d_basis.gpol(g1, g2)[source]¶
Return the G-Polynomial of
g_1andg_2.Let \(a_i t_i\) be \(LT(g_i)\), \(a = a_i*c_i + a_j*c_j\) with \(a = GCD(a_i,a_j)\), and \(s_i = t/t_i\) with \(t = LCM(t_i,t_j)\). Then the G-Polynomial is defined as: \(c_1s_1g_1 - c_2s_2g_2\).
INPUT:
g1– polynomialg2– polynomial
EXAMPLES:
sage: from sage.rings.polynomial.toy_d_basis import gpol sage: P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='lex') sage: f = x^2 - 1 sage: g = 2*x*y - z sage: gpol(f,g) x^2*y - y
>>> from sage.all import * >>> from sage.rings.polynomial.toy_d_basis import gpol >>> P = PolynomialRing(IntegerRing(), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = x**Integer(2) - Integer(1) >>> g = Integer(2)*x*y - z >>> gpol(f,g) x^2*y - y
from sage.rings.polynomial.toy_d_basis import gpol P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='lex') f = x^2 - 1 g = 2*x*y - z gpol(f,g)
- sage.rings.polynomial.toy_d_basis.select(P)[source]¶
The normal selection strategy.
INPUT:
P– list of critical pairs
OUTPUT: an element of P
EXAMPLES:
sage: from sage.rings.polynomial.toy_d_basis import select sage: A.<x,y> = PolynomialRing(ZZ, 2) sage: f = -y^2 - y + x^3 + 7*x + 1 sage: fx = f.derivative(x) sage: fy = f.derivative(y) sage: G = [f, fx, fy] sage: B = set((f1, f2) for f1 in G for f2 in G if f1 != f2) sage: select(B) (-2*y - 1, 3*x^2 + 7)
>>> from sage.all import * >>> from sage.rings.polynomial.toy_d_basis import select >>> A = PolynomialRing(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> f = -y**Integer(2) - y + x**Integer(3) + Integer(7)*x + Integer(1) >>> fx = f.derivative(x) >>> fy = f.derivative(y) >>> G = [f, fx, fy] >>> B = set((f1, f2) for f1 in G for f2 in G if f1 != f2) >>> select(B) (-2*y - 1, 3*x^2 + 7)
from sage.rings.polynomial.toy_d_basis import select A.<x,y> = PolynomialRing(ZZ, 2) f = -y^2 - y + x^3 + 7*x + 1 fx = f.derivative(x) fy = f.derivative(y) G = [f, fx, fy] B = set((f1, f2) for f1 in G for f2 in G if f1 != f2) select(B)
- sage.rings.polynomial.toy_d_basis.spol(g1, g2)[source]¶
Return the S-Polynomial of
g_1andg_2.Let \(a_i t_i\) be \(LT(g_i)\), \(b_i = a/a_i\) with \(a = LCM(a_i,a_j)\), and \(s_i = t/t_i\) with \(t = LCM(t_i,t_j)\). Then the S-Polynomial is defined as: \(b_1s_1g_1 - b_2s_2g_2\).
INPUT:
g1– polynomialg2– polynomial
EXAMPLES:
sage: from sage.rings.polynomial.toy_d_basis import spol sage: P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='lex') sage: f = x^2 - 1 sage: g = 2*x*y - z sage: spol(f,g) x*z - 2*y
>>> from sage.all import * >>> from sage.rings.polynomial.toy_d_basis import spol >>> P = PolynomialRing(IntegerRing(), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = x**Integer(2) - Integer(1) >>> g = Integer(2)*x*y - z >>> spol(f,g) x*z - 2*y
from sage.rings.polynomial.toy_d_basis import spol P.<x, y, z> = PolynomialRing(IntegerRing(), 3, order='lex') f = x^2 - 1 g = 2*x*y - z spol(f,g)
- sage.rings.polynomial.toy_d_basis.update(G, B, h)[source]¶
Update
Gusing the list of critical pairsBand the polynomialhas presented in [BW1993], page 230. For this, Buchberger’s first and second criterion are tested.This function uses the Gebauer-Moeller Installation.
INPUT:
G– an intermediate Groebner basisB– list of critical pairsh– a polynomial
OUTPUT:
G,BwhereGandBare updatedEXAMPLES:
sage: from sage.rings.polynomial.toy_d_basis import update sage: A.<x,y> = PolynomialRing(ZZ, 2) sage: G = set([3*x^2 + 7, 2*y + 1, x^3 - y^2 + 7*x - y + 1]) sage: B = set() sage: h = x^2*y - x^2 + y - 3 sage: update(G,B,h) ({2*y + 1, 3*x^2 + 7, x^2*y - x^2 + y - 3, x^3 - y^2 + 7*x - y + 1}, {(x^2*y - x^2 + y - 3, 2*y + 1), (x^2*y - x^2 + y - 3, 3*x^2 + 7), (x^2*y - x^2 + y - 3, x^3 - y^2 + 7*x - y + 1)})
>>> from sage.all import * >>> from sage.rings.polynomial.toy_d_basis import update >>> A = PolynomialRing(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> G = set([Integer(3)*x**Integer(2) + Integer(7), Integer(2)*y + Integer(1), x**Integer(3) - y**Integer(2) + Integer(7)*x - y + Integer(1)]) >>> B = set() >>> h = x**Integer(2)*y - x**Integer(2) + y - Integer(3) >>> update(G,B,h) ({2*y + 1, 3*x^2 + 7, x^2*y - x^2 + y - 3, x^3 - y^2 + 7*x - y + 1}, {(x^2*y - x^2 + y - 3, 2*y + 1), (x^2*y - x^2 + y - 3, 3*x^2 + 7), (x^2*y - x^2 + y - 3, x^3 - y^2 + 7*x - y + 1)})
from sage.rings.polynomial.toy_d_basis import update A.<x,y> = PolynomialRing(ZZ, 2) G = set([3*x^2 + 7, 2*y + 1, x^3 - y^2 + 7*x - y + 1]) B = set() h = x^2*y - x^2 + y - 3 update(G,B,h)