-
Notifications
You must be signed in to change notification settings - Fork 1
/
Cyclotomic.m2
235 lines (212 loc) · 7.17 KB
/
Cyclotomic.m2
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
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
-- -*- coding: utf-8 -*-
-- cyclotomic.m2
--
-- Copyright (C) 2009-12 Thomas Kahle <[email protected]>
--
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
--
-- This program is free software; you can redistribute it and/or modify
-- it under the terms of the GNU General Public License as published by
-- the Free Software Foundation; either version 2 of the License, or (at
-- your option) any later version.
--
-- This program is distributed in the hope that it will be useful, but
-- WITHOUT ANY WARRANTY; without even the implied warranty of
-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
-- General Public License for more details.
--
-- You should have received a copy of the GNU General Public License along
-- with this program; if not, write to the Free Software Foundation, Inc.,
-- 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.
--
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
newPackage(
"Cyclotomic",
Version => "1.0",
Date => "Februar 2012",
Authors => {{Name => "Thomas Kahle",
Email => "[email protected]",
HomePage => "http://thomas-kahle.de"}},
Headline => "Routines for cyclotomic fields",
Reload=>true
)
export {"cyclotomicField",
"cyclotomicPoly",
"findRootPower",
"joinCyclotomic"
}
ww := getSymbol "ww"
-- We use memoize to get physically the same cyclotomic field again and again!
cf = (i) -> (
Q := QQ(monoid [ww_i]);
toField ((Q) / cyclotomicPoly (i, Q_0)))
cyclotomicField = memoize cf
--- Here is an example of how to map coefficient rings to not forget it
-- restart
-- F = frac (QQ[a,b]/ideal(a^2-b))
-- G = frac (QQ[x,y]/ideal(x^2-y))
--
-- R = F[t]
-- S = G[t]
--
-- use R
-- I = ideal (t-a-b)
-- use S
-- J = ideal (t+x)
--
-- use S
-- f = map (S,R,{t,x,y^2})
-- f(I)
-- We apply this to join cyclotomic fields
joinCyclotomic = li -> (
-- This function joins a list of ideals in a polynomial ring over a smallest common cf. Input should consist of
-- ideals in polynomial rings over the same variables, but with possible different cyclotomic coefficient field
-- Find root powers:
lc := for l in li list findRootPower ring l;
leastcm := lcm for l in lc list if l>2 then l else 1;
--Check for QQ
if leastcm < 3 then return li;
F := cyclotomicField leastcm;
-- Here we use the assumptions that all rings have the same generators
-- This should contain the variables of polynomials
ge := gens ring li#0;
S := F(monoid [ge]);
li2 := {}; ww:=F_0; local f;
for i in 0..#li-1 do (
if lc#i == 2 then (
-- rational coefficients: just map
li2 = li2 | {sub (li#i,S)};
)
else (
-- Was cyclotomic: need to find image of ww in new ring!
f = map (S, ring li#i , (gens S) |{ww^(leastcm//lc#i)});
li2 = li2 | { f li#i };
);
);
li2)
cyclotomicPoly = (i,v) -> (
-- returns the i-th cyclotomic polynomial in variable v.
-- v must be a variable a priori
v = value v;
if i <= 0 then error "the input should be > 0.";
if i==1 then return v-1 ;
mini := v^i -1;
-- dividing out the first cylcotomic polynomial
-- (with result a polynomial)
mini = (flatten entries syz matrix {{mini ,(v-1)}})#1;
-- i is prime:
if isPrime i then return mini / (leadCoefficient mini);
-- i is not prime:
-- find the divisors:
for f in toList (2..i//2) do (
-- check for factor
if i%f == 0 then (
fac := cyclotomicPoly (f,v);
-- division with result in polynomial ring:
mini = (flatten entries syz matrix {{mini,fac}})#1;
)
);
--make sure the leading coefficient is one.
mini / leadCoefficient(mini))
findRootPower = R -> (
-- Finds the power of the adjoined root of unity in the
-- coefficient ring of R by just exponentiating.
-- Returns '2' if the input was a polynomial ring over QQ
r := 0;
F := coefficientRing R;
fieldgens := (K,F) -> if K === F then {} else for x in gens last F.baseRings list promote(x,F);
g := fieldgens (QQ,F);
if #g == 0 then return 2;
if #g > 1 then error "The coefficient field has more than one generator";
g = value (g#0);
gg := g; -- the generator
while not 1_F == gg do (
r = r+1;
gg = gg *g;
);
if r<2 then return 2
else r+1)
-- End of source code ---
beginDocumentation()
document {
Key => Cyclotomic,
Headline => "a package for cyclotomic fields",
EM "Cyclotomic", " is a package for cyclotomic fields. It is used in
to construct extensions of the coefficient field during binomial
primary decomposition using the package ", TO "Binomials::Binomials", "."
}
document {
Key => {cyclotomicField},
Headline => "Cyclotomic Field Construction",
Usage => "cyclotomicField (i)",
Inputs => {
"i" => { "an integer, the power of the root to be adjoined."}},
Outputs => {
"S" => {"A cyclotomic field with $1^(1/i)$ adjoined"} },
EXAMPLE {
"S = cyclotomicField (5)",
"isField S",
"r = S_0",
"(r^9, r^10, r^11)",
"T = S[x,y]",
"I = ideal (x-r)",
"dim I"
},
Caveat => {"Strange things can happen with the reduction of the coefficients.", " In M2 v <= 1.2 dimension is off by one."},
SeeAlso => cyclotomicPoly
}
document {
Key => {cyclotomicPoly},
Headline => "Cyclotomic Polynomial",
Usage => "cyclotomicPoly (i,v)",
Inputs => {
"i" => { "an integer, the power of a root of unity."},
"v" => { "a variable name in which the polynomial is returned."} },
Outputs => {
"f" => {"The minimal polynomial of the i-th root of unity."} },
EXAMPLE {
"R = QQ[ww]",
"f = cyclotomicPoly (6,ww)",
},
SeeAlso => cyclotomicField
}
document {
Key => {findRootPower},
Headline => "Find the order of the root in the coefficients of polynomial ring over a cyclotomic field",
Usage => "findRootPower R",
Inputs => {
"R" => { "a polynomial ring over a cyclotomic field"}},
Outputs => {
"i" => {"The order of the adjoined root of unity, or 2 if the coefficient field is QQ"} },
EXAMPLE {
"S = cyclotomicField 5",
"T = S[x,y]",
"findRootPower T"
},
SeeAlso => cyclotomicField
}
document {
Key => {joinCyclotomic},
Headline => "Join ideals in polynomial rings over different cyclotomic fields",
Usage => "joinCyclotomic l",
Inputs => {
"li" => { "a list of ideals in polynomial rings over cyclotomic fields"}},
Outputs => {
"li2" => {"The list of ideals in a common ring."} },
EXAMPLE {
"F = cyclotomicField 3; G = cyclotomicField 4;",
"R = F[t]; I = ideal (t-F_0^2)",
"S = G[t]; J = ideal (t^2-G_0)",
"joinCyclotomic {I,J}"
},
SeeAlso => cyclotomicField
}
TEST ///
F = cyclotomicField 3
G = cyclotomicField 4
R = F[t]
I = ideal (t-F_0^2)
S = G[t]
J = ideal (t^2-G_0)
assert (findRootPower coefficientRing ring (joinCyclotomic {I,J})#0 == 12)
///