source: CGBLisp/examples/pilgrim3.lisp@ 1

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From pilgrim@polygon.math.cornell.edu  Tue Oct 21 12:07:20 1997
Return-Path: pilgrim@polygon.math.cornell.edu
Date: Tue, 21 Oct 1997 15:06:59 -0400 (EDT)
From: "Kevin M. Pilgrim - Math H.C. Wang Visiting Prof." <pilgrim@math.cornell.edu>
X-Sender: pilgrim@polygon
To: rychlik@math.arizona.edu
Subject: A test for grobner bases algorithms
Mime-Version: 1.0
Content-Type: TEXT/PLAIN; charset=US-ASCII

Hi, 

Below are some sample equations I looked at this summer.

Here's the setup:

A general degree two rational map is (generically) conjugate in at 
most six ways to a map of the form 

        f_ab = z(az^2 + bz + a) = 1/ (a (z+1/z) + b), a nonzero.

as you can easily check using algebra.  The critical points are at 
+/- 1 and infinity maps to zero, which is fixed.  
Such a map has three fixed points whose multipliers are, say, m1, m2, m3.
Let s1=m1+m2+m3 and s2=m1*m2 + m1*m3 + m2*m3 be the first two 
elementary symmetric functions of these multipliers.   From work of 
Milnor (generalized by Silverman) one knows that s1 and s2 give canonical
coordinates for the moduli space of degree two rational maps, i.e. the space
of rational maps of degree two modulo moebius conjugation is biholomorphic
to C^2 with these canonical coordinates s1 and s2.

One can easily check that 

        a*s1-4*a^2-1+b^2+2*a = 0                                
and                                                             (E1)
        a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2 = 0.           

Now, let s denote the ith elementary symmetric function of the multipliers 
of the points of period n.  From work of Silverman, 

        s is a polynomial with integer coefficients in s1 and s2.

PROBLEM:  Calculate s in terms of i and n.

Even for n=3 and i=1 this is not easy to do using "brute force" 
symbolic computation.  But somehow there ought to be a way to do 
this using this approach.  

Case n=3 and i=1:  
-----------------

A degree two rational map has two three-cycles, generically.  
Let s = s(3,1) denote the sum of the multipliers of these two three-cycles.
We want to find s in terms of s1 and s2.

Let 
        z00 -> z10 -> z20 -> z00 
and                                                             (E2)
        z01 -> z11 -> z21 -> z01

denote the points in the two three-cycles (the arrows denote application
of f_ab).  Then 

        s = f'(z00)*f'(z10)*f'(z20)+f'(z01)*f'(z11)*f'(z21)     (E3)

and s is the sum of these two quantities and depends only on s1 and s2.

The conditions in (E2) and (E3) can be translated into polynomial
equations.  Adding in the equations (E1) gives us a system of poly.
equations.  Here are the details, in a maple-readable form:

variables:=[z21, z11, z01, z20, z10, z00, a, b, s1, s2, s];
equations:=
[a*s1-4*a^2-1+b^2+2*a, 
a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2, 
z00-z10*(a*z00^2+b*z00+a), z10-z20*(a*z10^2+b*z10+a), 
z20-z00*(a*z20^2+b*z20+a), z01-z11*(a*z01^2+b*z01+a), 
z11-z21*(a*z11^2+b*z11+a), z21-z01*(a*z21^2+b*z21+a), 
s-a^3*(1-z00^2)*(1-z10^2)*(1-z20^2)-a^3*(1-z01^2)*(1-z11^2)*(1-z21^2)];

There are, however, some degeneracy constraints:  we may not allow the
coefficient a to be zero.  Also, there will be solutions to the equations
where e.g. z00=z01, z10=z11, z20=z21 which we must discard.  So we have
the following quantities which must be nonzero: 

(D1)    a 
 
(D2)    z00-z10, z00-z20, z10-z20,
        z01-z11, z01-z21, z11-z21,

        (where points in the same cycle collide)
and

(D3)    z00-z01, z00-z11, z00-z21,
        z10-z01, z10-z11, z10-z21,
        z20-z01, z20-z11, z20-z21
        
        (where points in different cycles collide).

But in fact one can reduce this set of 1 + 6 + 9 = 16 
degeneracy conditions somewhat using the fact that the 
zij's are period of the prime period 3 to:

degs:=[a, z00-z10, z01-z11, z00-z01, z00-z11, z00-z21];

which is a great improvement.  

Thus our problem is:

        Solve "equations" in "variables" subject to the 
        constraint that "degs" are nonzero, and eliminate
        all the variables save s,s1,s2.

An alternative approach, which you mentioned, is to first 
omit consideration of the degeneracy equations.  

I'm curious to see how you would approach this problem
using what you called parameters.  I'd be grateful for
any input you could give.  

If this one looks too easy, I can send you the data for 
higher periods.  For e.g. n=4 there are 3 four-cycles and the 
degeneracies which can occur are more complicated.  

Another useful thing to have would be the equations in a,b, and 
the zij's defining the locus where the values of the zij's represent 
precisely the locations of the points in the two three-cycles
(i.e. calculate a basis for the ideal corresponding to the variety 
defined by (E2) subject to the constraint that the degeneracies (D1)-(D3)
are nonzero.  Even this looks hard.

If you're interested I can send some samples of the equations which 
arise when trying to find postcritically finite maps.  

I enjoyed your talk in Atlanta, and thank you for organizing a great
session and allowing me the opportunity to speak.

        Kevin M. Pilgrim        
        Dept. of Mathematics
        B-35 White Hall
        Cornell University
        Ithaca, NY 14853-7901
        (607) 255-5380 w
        (607) 256-1634 h, before 9PM
        pilgrim@math.cornell.edu

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(setf variables '(s1 s2 s)) 
(setf parameters '(w00 w10 w20 w01 w11 w21 r00 r10 r20 r01 r11 r21 z21 z11  z01 z20 z10 z00 a b)) 

(setf equations
"[ a*s1-4*a^2-1+b^2+2*a,
 a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2,
 s-a^3*r00*r10*r20-a^3*r01*r11*r21
]"
)

(setf green-list "[
 a*z00^2+b*z00+a-w00,
 a*z10^2+b*z10+a-w10,
 a*z20^2+b*z20+a-w20, 
 a*z01^2+b*z01+a-w01,
 a*z11^2+b*z11+a-w11,
 a*z21^2+b*z21+a-w21,
 z00-z10*w00,
 z10-z20*w10, 
 z20-z00*w20,
 z01-z11*w01, 
 z11-z21*w11,
 z21-z01*w21, 
 1-z00^2-r00,
 1-z10^2-r10,
 1-z20^2-r20,
 1-z01^2-r01,
 1-z11^2-r11,
 1-z21^2-r21
]")

(setf red-list "[a
 z00-z10
 z01-z11
 z00-z01
 z00-z11
 z00-z21]")

(setf order #'lex>)

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(string-grobner green-list parameters :order order)
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(setf eqns (string-grobner equations variables :order order))
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(string-ideal-polysaturation-1
 equations
 red-list
 (append parameters variables)
 :order order
) 
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