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regSeqInIdeal -- a regular sequence contained in an ideal

Synopsis

Description

This method computes a regular sequence of length n contained in a given ideal I. It attempts to do so by first trying "sparse" combinations of the generators, i.e. elements which are either generators or sums of two generators. If a sparse regular sequence is not found, then dense combinations of generators will be tried.

If the length n is either unspecified or greater than the codimension of I then it is silently replaced with the codimension of I. The ideal I should be in a polynomial (or at least Cohen-Macaulay) ring, so that codim I = grade I.

i1 : R = QQ[x_0..x_7]

o1 = R

o1 : PolynomialRing
 
i2 : I = intersect(ideal(x_0,x_1,x_2,x_3), ideal(x_4,x_5,x_6,x_7), ideal(x_0,x_2,x_4,x_6), ideal(x_1,x_3,x_5,7))

o2 = ideal (x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,
             2 7   0 7   3 6   2 6   1 6   0 6   2 5   0 5   3 4   2 4   1 4 
     ------------------------------------------------------------------------
     x x )
      0 4

o2 : Ideal of R
 
i3 : elapsedTime regSeqInIdeal I
 -- 0.0345811 seconds elapsed

o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
             2 7   3 6    0 7   2 5    0 7   1 4    0 7

o3 : Ideal of R

If I is the unit ideal, then an ideal of variables of the ring is returned.

If the codimension of I is already known, then one can specify this, along with a time limit for each trial (normally this is taken from the length of time for computing codim I). This can result in a significant speedup: in the following example, codim I takes more than a minute to complete.

i4 : R = QQ[h,l,s,x,y,z]

o4 = R

o4 : PolynomialRing
 
i5 : I = ideal(h*l-l^2-4*l*s+h*y,h^2*s-6*l*s^3+h^2*z,x*h^2-l^2*s-h^3,h^8,l^8,s^8)

                   2                     3    2     2      3    2     2    8 
o5 = ideal (h*l - l  - 4l*s + h*y, - 6l*s  + h s + h z, - h  - l s + h x, h ,
     ------------------------------------------------------------------------
      8   8
     l , s )

o5 : Ideal of R
 
i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3

o6 = true
 
i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
 -- 0.00605052 seconds elapsed

                   2                3    2     2    8    3    2     2
o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)

o7 : Ideal of R

See also

Ways to use regSeqInIdeal :

For the programmer

The object regSeqInIdeal is a method function with options.