relationType -- Given an ideal in a ring this computes the maximum degree, of the new variables, of the minimal generators of the defining ideal of the associated Rees algebra.

Synopsis

Usage:
n = relationType(I, L)
n = relationType(I, J)
n = relationType(R,J)
Inputs:
- I, an ideal, The ideal defining the base ring R.
- L, a list, The list of generators of the ideal J we are forming the Rees algebra of.
- R, a ring, The base ring.
- J, an ideal, The ideal we are forming the Rees algebra of.
Optional inputs:
- Strategy => ..., -- Determines the desired Strategy in each function.
- Verbose => ..., -- generate informative output
Outputs:
- n, an integer, The maximum degree of the generators of the defining ideal of the Rees algebra.

Description

Suppose ( g₁, ..., g_m ) = J ⊆R is an ideal in a ring R. We form the Rees algebra R[Jt] = R[Y₁, ..., Y_m]/K where the Y_i map to the g_i. This function returns the maximum Y-degree of the generators of K. For more information, see page 22 of Vasconcelos, Rees algebras, multiplicities, algorithms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.

i1 : R = QQ[x_0..x_8];

i2 : M = genericMatrix(R,x_0,3,3)

o2 = | x_0 x_3 x_6 |
     | x_1 x_4 x_7 |
     | x_2 x_5 x_8 |

             3       3
o2 : Matrix R  <--- R

i3 : J = minors (2,M)

o3 = ideal (- x x  + x x , - x x  + x x , - x x  + x x , - x x  + x x , -
               1 3    0 4     2 3    0 5     2 4    1 5     1 6    0 7   
     ------------------------------------------------------------------------
     x x  + x x , - x x  + x x , - x x  + x x , - x x  + x x , - x x  + x x )
      2 6    0 8     2 7    1 8     4 6    3 7     5 6    3 8     5 7    4 8

o3 : Ideal of R

i4 : relationType(R,J)
blowUpIdeals computed.

o4 = 1

Ways to use `relationType` :

relationType(Ideal,BasicList)
relationType(Ideal,Ideal)
relationType(Ring,Ideal)

relationType -- Given an ideal in a ring this computes the maximum degree, of the new variables, of the minimal generators of the defining ideal of the associated Rees algebra.

Synopsis

Description

Ways to use relationType :

Ways to use `relationType` :