degree(Module)

Synopsis

• Function: degree

• Usage:

  degree M

• Inputs:
  • M, a module, over a polynomial ring or quotient of a polynomial ring, over a field k
• Outputs:
  • an integer, the degree of M

Description

If M is finite dimensional over k, the degree of M is its dimension over k. Otherwise, the degree of M is the multiplicity of M, i.e., the integer d such that the Hilbert polynomial of M has the form z |--> d z^e/e! + lower terms in z.

i1 : R = ZZ/101[t,x,y,z];

i2 : degree (R^1 / (ideal vars R)^6)

o2 = 126

i3 : degree minors_2 matrix {{t,x,y},{x,y,z}}

o3 = 3

The algorithm computes the poincare polynomial of M, divides it by 1-T as often as possible, then evaluates it at T=1. When the module has finite length, the result is the Hilbert series evaluated at 1, that is the dimension over the ground field, which for a graded (homogeneous) is the same as the length.

Caveat

See also

• hilbertPolynomial -- compute the Hilbert polynomial
• isHomogeneous -- whether something is homogeneous (graded)