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degree(Module)

Synopsis

Description

We assume that M is a graded (homogeneous) module over a singly graded polynomal ring or a quotient of a polynomial ring, over a field k.

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 hilbertSeries of M (as a rational function), divides both numerator and denominator by 1-T as often as possible, then evaluates both at T=1 and returns the resulting quotient as a (possibly rational) number. When the module has finite length, then the rational function is a polynomial, and evaluating it at 1 returns the dimension over the ground field, which for a graded (homogenous) is the same as the length.

Caveat

If the base ring is ZZ, or the module is not homogeneous, it is likely that the answer is not what you would expect. Similarly, if the degrees of the variables are not all {1}, then the answer is harder to interpret. See heft vectors and multidegree.

See also