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
|
--- status: Draft
--- author(s): Decker
--- notes:
document {
Key => {irreducibleCharacteristicSeries,(irreducibleCharacteristicSeries,Ideal)},
Headline => "irreducible characteristic series of an ideal",
Usage => "(ics,p) = irreducibleCharacteristicSeries I",
Inputs => {"I" => Ideal },
Outputs => {
"ics" => List => {"a list of matrices, representing an irreducible characteristic series for ", TT "I"},
"p" => RingMap => {"an isomorphism from the ring of ", TT "I", " to the ring of the characteristic series.
The ring retains the names and degrees of the variables, but reorders the variables and uses a
default monomial ordering."
}
},
PARA {
TEX ///
As we see in the example below, an irreducible characteristic series
for $I$ consists of a collection of triangular sets. Here,
given a polynomial $f$, write $lvar(f)$ for the
largest variable appearing in $f$ (with respect to the lexicographic order).
In the example, $lvar(-y w+x^2) = y$ . A triangular set consists
of polynomials $f_1,\dots,f_r$ such that $lvar(f_1)< \dots < lvar(f_r)$.
In the example, $lvar(-x*y^2+z^3) = x < w = lvar(-w*y+z^2)$ . If
$T_1,\dots,T_s$ form an irreducible characteristic series for $I$ , and if
$J_i$ is the ideal generated by the largest variables of the
elements of $T_i$ , then the algebraic set $V(I)$
defined by $I$ is the union of the sets $V(T_i) \setminus V(I_i)$, for $i=1,\dots,s$.
The minimal associated primes of $I$ can thus be recovered from the
irreducible characteristic series by saturation and by throwing away superfluous primes.
///,
"This is done by ", TO "MinimalPrimes :: minimalPrimes", ", which uses this routine."
},
EXAMPLE lines ///
R = QQ[w,x,y,z];
(L,p) = irreducibleCharacteristicSeries ideal(x^2-y*w,x^3-z*w^2)
apply(L, m -> p m)
p^-1
///,
SeeAlso => {"MinimalPrimes :: minimalPrimes"}
}
|
