1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
|
--- status: converted November 2021
--- author(s):
--- notes: heft is defined in hilbert.m2
doc ///
Node
Key
"heft vectors"
Description
Text
A @EM "heft vector"@ for a polynomial ring is a vector with integer entries, of the same
length as the degree vectors of the variables of the ring, whose dot product with each of
them is (strictly) positive. Unless one is specified explicitly, then a good one will be
found automatically. The heft vector is used in various internal algorithms, such as the
one in @TO "basis"@, as a way of organizing the sequence of steps, proceeding incrementally
to larger values of the dot product of the degree of a monomial with the heft vector.
Example
R = QQ[a..d];
degrees R
heft R
S = QQ[a..d, DegreeRank => 4];
degrees S
heft S
T = QQ[a,b, Degrees => {1,-1}]
degrees T
heft T
U = QQ[a..d, Degrees => {{2,0}, {1,-1}, {0,-2}, {-1,-3}}]
degrees U
heft U
Text
The heft vector, multiplied by -1, is used as the weight vector in the monomial ordering of
the degrees ring, and the @EM "order"@ of the series expansions of the Hilbert series refers
to the weight formed with respect to that weight vector.
Example
hilbertSeries U
describe ring numerator oo
hilbertSeries(U, Order => 8)
Text
The heft vector is used in the computation of degrees of modules over a polynomial ring @TT "R"@,
because it gives a homomorphism from the degrees ring of @TT "R"@ to the Laurent polynomial
ring in one variable @TT "T"@ that sends monomials corresponding to the degrees of variables
of @TT "R"@ to positive powers of @TT "T"@. See @TO (degree, Module)@.
Example
R = QQ[x,y, Heft => {3}];
degree ideal x
SeeAlso
[monoid, Heft]
degreesRing
multidegree
[hilbertSeries, Order]
Subnodes
heft
findHeft
Node
Key
heft
(heft, Ring)
(heft, Monoid)
(heft, PolynomialRing)
(heft, QuotientRing)
Headline
heft vector of ring or monoid
Usage
heft R
Inputs
R:{Ring,Monoid}
Outputs
:List
a weight co-vector of integers with length the same as @TO2 {degreeLength, "degree length"}@ of @TT "R"@, or
:Nothing
@TO "null"@ if there is no heft vector.
Description
Text
@TT "heft"@ returns the @TO2 {"heft vectors", "heft vector"}@ in use for @TT "R"@,
Example
S = QQ[a..d, DegreeRank => 4];
degrees S
heft S
SeeAlso
findHeft
Node
Key
findHeft
(findHeft, List)
[findHeft, DegreeRank]
Headline
find a heft vector for a list of degrees
Usage
findHeft(degs, DegreeRank => r)
Inputs
degs:List
of multi-degrees, each of which is a list of integers of fixed length {\tt r}
DegreeRank=>ZZ
Outputs
:List
of integers of length {\tt r}, or
:Nothing
@TO "null"@, if there is none
Description
Text
@TT "findHeft"@ returns a weight co-vector whose dot product with each member of @TT "degs"@ is strictly positive.
Example
findHeft({{-1,0}, {2,1}}, DegreeRank => 2)
findHeft({}, DegreeRank => 0)
SeeAlso
heft
///
|
