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 [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].

EXAMPLE:

sage: 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()
[1]

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
[x - 2020, y - 11313, 22627]

sage: gb[-1].factor()
11^3 * 17

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: 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

To compute modulo 4, we can add the generator 4 to our basis.:

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]

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)
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()
[1]

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: d_basis(I) # random -- waiting on upstream singular fixes at #6051
[x + 170269749119, y + 2149906854, z + ..., 282687803443]

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()
[x + 19, y + 48, z - 33]

sage: I.change_ring( P.change_ring( GF(103) ) ).groebner_basis()
[x + 39, y + 8, z - 18]

sage: I.change_ring( P.change_ring( GF(27173681) ) ).groebner_basis()
[x - 536027, y + 3186055, z + 10380032]

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()
[1]

AUTHOR:

  • Martin Albrecht (2008-08): initial version
sage.rings.polynomial.toy_d_basis.LC(f)

x.__init__(...) initializes x; see help(type(x)) for signature

sage.rings.polynomial.toy_d_basis.LM(f)

x.__init__(...) initializes x; see help(type(x)) for signature

sage.rings.polynomial.toy_d_basis.d_basis(F, strat=True)

Return the \(d\)-basis for the Ideal F as defined in [BW93].

INPUT:

  • F - an ideal
  • strat - use update strategy (default: True)

EXAMPLE:

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
[x - 2020, y - 11313, 22627]
sage.rings.polynomial.toy_d_basis.gpol(g1, g2)

Return G-Polynomial of g_1 and g_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 - polynomial
  • g2 - polynomial

EXAMPLE:

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
sage.rings.polynomial.toy_d_basis.select(P)

The normal selection strategy.

INPUT:

  • P - a list of critical pairs
OUTPUT:
an element of P

EXAMPLE:

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(filter(lambda (x,y): x!=y, [(f1,f2) for f1 in G for f2 in G]))
sage: select(B)
(-2*y - 1, 3*x^2 + 7)
sage.rings.polynomial.toy_d_basis.spol(g1, g2)

Return S-Polynomial of g_1 and g_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 - polynomial
  • g2 - polynomial

EXAMPLE:

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
sage.rings.polynomial.toy_d_basis.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 uses the Gebauer-Moeller Installation.

INPUT:

  • G - an intermediate Groebner basis
  • B - a list of critical pairs
  • h - a polynomial
OUTPUT:
G,B where G and B are updated

EXAMPLE:

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)})

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