1 | #|
|
---|
2 | From pilgrim@polygon.math.cornell.edu Tue Oct 21 12:07:20 1997
|
---|
3 | Return-Path: pilgrim@polygon.math.cornell.edu
|
---|
4 | Date: Tue, 21 Oct 1997 15:06:59 -0400 (EDT)
|
---|
5 | From: "Kevin M. Pilgrim - Math H.C. Wang Visiting Prof." <pilgrim@math.cornell.edu>
|
---|
6 | X-Sender: pilgrim@polygon
|
---|
7 | To: rychlik@math.arizona.edu
|
---|
8 | Subject: A test for grobner bases algorithms
|
---|
9 | Mime-Version: 1.0
|
---|
10 | Content-Type: TEXT/PLAIN; charset=US-ASCII
|
---|
11 |
|
---|
12 | Hi,
|
---|
13 |
|
---|
14 | Below are some sample equations I looked at this summer.
|
---|
15 |
|
---|
16 | Here's the setup:
|
---|
17 |
|
---|
18 | A general degree two rational map is (generically) conjugate in at
|
---|
19 | most six ways to a map of the form
|
---|
20 |
|
---|
21 | f_ab = z(az^2 + bz + a) = 1/ (a (z+1/z) + b), a nonzero.
|
---|
22 |
|
---|
23 | as you can easily check using algebra. The critical points are at
|
---|
24 | +/- 1 and infinity maps to zero, which is fixed.
|
---|
25 | Such a map has three fixed points whose multipliers are, say, m1, m2, m3.
|
---|
26 | Let s1=m1+m2+m3 and s2=m1*m2 + m1*m3 + m2*m3 be the first two
|
---|
27 | elementary symmetric functions of these multipliers. From work of
|
---|
28 | Milnor (generalized by Silverman) one knows that s1 and s2 give canonical
|
---|
29 | coordinates for the moduli space of degree two rational maps, i.e. the space
|
---|
30 | of rational maps of degree two modulo moebius conjugation is biholomorphic
|
---|
31 | to C^2 with these canonical coordinates s1 and s2.
|
---|
32 |
|
---|
33 | One can easily check that
|
---|
34 |
|
---|
35 | a*s1-4*a^2-1+b^2+2*a = 0
|
---|
36 | and (E1)
|
---|
37 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2 = 0.
|
---|
38 |
|
---|
39 | Now, let s denote the ith elementary symmetric function of the multipliers
|
---|
40 | of the points of period n. From work of Silverman,
|
---|
41 |
|
---|
42 | s is a polynomial with integer coefficients in s1 and s2.
|
---|
43 |
|
---|
44 | PROBLEM: Calculate s in terms of i and n.
|
---|
45 |
|
---|
46 | Even for n=3 and i=1 this is not easy to do using "brute force"
|
---|
47 | symbolic computation. But somehow there ought to be a way to do
|
---|
48 | this using this approach.
|
---|
49 |
|
---|
50 | Case n=3 and i=1:
|
---|
51 | -----------------
|
---|
52 |
|
---|
53 | A degree two rational map has two three-cycles, generically.
|
---|
54 | Let s = s(3,1) denote the sum of the multipliers of these two three-cycles.
|
---|
55 | We want to find s in terms of s1 and s2.
|
---|
56 |
|
---|
57 | Let
|
---|
58 | z00 -> z10 -> z20 -> z00
|
---|
59 | and (E2)
|
---|
60 | z01 -> z11 -> z21 -> z01
|
---|
61 |
|
---|
62 | denote the points in the two three-cycles (the arrows denote application
|
---|
63 | of f_ab). Then
|
---|
64 |
|
---|
65 | s = f'(z00)*f'(z10)*f'(z20)+f'(z01)*f'(z11)*f'(z21) (E3)
|
---|
66 |
|
---|
67 | and s is the sum of these two quantities and depends only on s1 and s2.
|
---|
68 |
|
---|
69 | The conditions in (E2) and (E3) can be translated into polynomial
|
---|
70 | equations. Adding in the equations (E1) gives us a system of poly.
|
---|
71 | equations. Here are the details, in a maple-readable form:
|
---|
72 |
|
---|
73 | variables:=[z21, z11, z01, z20, z10, z00, a, b, s1, s2, s];
|
---|
74 | equations:=
|
---|
75 | [a*s1-4*a^2-1+b^2+2*a,
|
---|
76 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2,
|
---|
77 | z00-z10*(a*z00^2+b*z00+a), z10-z20*(a*z10^2+b*z10+a),
|
---|
78 | z20-z00*(a*z20^2+b*z20+a), z01-z11*(a*z01^2+b*z01+a),
|
---|
79 | z11-z21*(a*z11^2+b*z11+a), z21-z01*(a*z21^2+b*z21+a),
|
---|
80 | s-a^3*(1-z00^2)*(1-z10^2)*(1-z20^2)-a^3*(1-z01^2)*(1-z11^2)*(1-z21^2)];
|
---|
81 |
|
---|
82 | There are, however, some degeneracy constraints: we may not allow the
|
---|
83 | coefficient a to be zero. Also, there will be solutions to the equations
|
---|
84 | where e.g. z00=z01, z10=z11, z20=z21 which we must discard. So we have
|
---|
85 | the following quantities which must be nonzero:
|
---|
86 |
|
---|
87 | (D1) a
|
---|
88 |
|
---|
89 | (D2) z00-z10, z00-z20, z10-z20,
|
---|
90 | z01-z11, z01-z21, z11-z21,
|
---|
91 |
|
---|
92 | (where points in the same cycle collide)
|
---|
93 | and
|
---|
94 |
|
---|
95 | (D3) z00-z01, z00-z11, z00-z21,
|
---|
96 | z10-z01, z10-z11, z10-z21,
|
---|
97 | z20-z01, z20-z11, z20-z21
|
---|
98 |
|
---|
99 | (where points in different cycles collide).
|
---|
100 |
|
---|
101 | But in fact one can reduce this set of 1 + 6 + 9 = 16
|
---|
102 | degeneracy conditions somewhat using the fact that the
|
---|
103 | zij's are period of the prime period 3 to:
|
---|
104 |
|
---|
105 | degs:=[a, z00-z10, z01-z11, z00-z01, z00-z11, z00-z21];
|
---|
106 |
|
---|
107 | which is a great improvement.
|
---|
108 |
|
---|
109 | Thus our problem is:
|
---|
110 |
|
---|
111 | Solve "equations" in "variables" subject to the
|
---|
112 | constraint that "degs" are nonzero, and eliminate
|
---|
113 | all the variables save s,s1,s2.
|
---|
114 |
|
---|
115 | An alternative approach, which you mentioned, is to first
|
---|
116 | omit consideration of the degeneracy equations.
|
---|
117 |
|
---|
118 | I'm curious to see how you would approach this problem
|
---|
119 | using what you called parameters. I'd be grateful for
|
---|
120 | any input you could give.
|
---|
121 |
|
---|
122 | If this one looks too easy, I can send you the data for
|
---|
123 | higher periods. For e.g. n=4 there are 3 four-cycles and the
|
---|
124 | degeneracies which can occur are more complicated.
|
---|
125 |
|
---|
126 | Another useful thing to have would be the equations in a,b, and
|
---|
127 | the zij's defining the locus where the values of the zij's represent
|
---|
128 | precisely the locations of the points in the two three-cycles
|
---|
129 | (i.e. calculate a basis for the ideal corresponding to the variety
|
---|
130 | defined by (E2) subject to the constraint that the degeneracies (D1)-(D3)
|
---|
131 | are nonzero. Even this looks hard.
|
---|
132 |
|
---|
133 | If you're interested I can send some samples of the equations which
|
---|
134 | arise when trying to find postcritically finite maps.
|
---|
135 |
|
---|
136 | I enjoyed your talk in Atlanta, and thank you for organizing a great
|
---|
137 | session and allowing me the opportunity to speak.
|
---|
138 |
|
---|
139 | Kevin M. Pilgrim
|
---|
140 | Dept. of Mathematics
|
---|
141 | B-35 White Hall
|
---|
142 | Cornell University
|
---|
143 | Ithaca, NY 14853-7901
|
---|
144 | (607) 255-5380 w
|
---|
145 | (607) 256-1634 h, before 9PM
|
---|
146 | pilgrim@math.cornell.edu
|
---|
147 |
|
---|
148 | |#
|
---|
149 |
|
---|
150 |
|
---|
151 | (setf variables '(s1 s2 s))
|
---|
152 | (setf parameters '(w00 w10 w20 w01 w11 w21 r00 r10 r20 r01 r11 r21 z21 z11 z01 z20 z10 z00 a b))
|
---|
153 |
|
---|
154 | (setf equations
|
---|
155 | "[ a*s1-4*a^2-1+b^2+2*a,
|
---|
156 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2,
|
---|
157 | s-a^3*r00*r10*r20-a^3*r01*r11*r21
|
---|
158 | ]"
|
---|
159 | )
|
---|
160 |
|
---|
161 | (setf green-list "[
|
---|
162 | a*z00^2+b*z00+a-w00,
|
---|
163 | a*z10^2+b*z10+a-w10,
|
---|
164 | a*z20^2+b*z20+a-w20,
|
---|
165 | a*z01^2+b*z01+a-w01,
|
---|
166 | a*z11^2+b*z11+a-w11,
|
---|
167 | a*z21^2+b*z21+a-w21,
|
---|
168 | z00-z10*w00,
|
---|
169 | z10-z20*w10,
|
---|
170 | z20-z00*w20,
|
---|
171 | z01-z11*w01,
|
---|
172 | z11-z21*w11,
|
---|
173 | z21-z01*w21,
|
---|
174 | 1-z00^2-r00,
|
---|
175 | 1-z10^2-r10,
|
---|
176 | 1-z20^2-r20,
|
---|
177 | 1-z01^2-r01,
|
---|
178 | 1-z11^2-r11,
|
---|
179 | 1-z21^2-r21
|
---|
180 | ]")
|
---|
181 |
|
---|
182 | (setf red-list "[a
|
---|
183 | z00-z10
|
---|
184 | z01-z11
|
---|
185 | z00-z01
|
---|
186 | z00-z11
|
---|
187 | z00-z21]")
|
---|
188 |
|
---|
189 | (setf order #'lex>)
|
---|
190 |
|
---|
191 | #|
|
---|
192 | (string-grobner green-list parameters :order order)
|
---|
193 | |#
|
---|
194 |
|
---|
195 | #|
|
---|
196 | (setf eqns (string-grobner equations variables :order order))
|
---|
197 | |#
|
---|
198 |
|
---|
199 |
|
---|
200 | #|
|
---|
201 | (string-ideal-polysaturation-1
|
---|
202 | equations
|
---|
203 | red-list
|
---|
204 | (append parameters variables)
|
---|
205 | :order order
|
---|
206 | )
|
---|
207 | |#
|
---|
208 |
|
---|