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| 2 | From pilgrim@polygon.math.cornell.edu Tue Oct 21 12:07:20 1997
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| 3 | Return-Path: pilgrim@polygon.math.cornell.edu
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| 4 | Date: Tue, 21 Oct 1997 15:06:59 -0400 (EDT)
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| 5 | From: "Kevin M. Pilgrim - Math H.C. Wang Visiting Prof." <pilgrim@math.cornell.edu>
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| 6 | X-Sender: pilgrim@polygon
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| 7 | To: rychlik@math.arizona.edu
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| 8 | Subject: A test for grobner bases algorithms
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| 9 | Mime-Version: 1.0
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| 10 | Content-Type: TEXT/PLAIN; charset=US-ASCII
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| 11 |
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| 12 | Hi,
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| 13 |
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| 14 | Below are some sample equations I looked at this summer.
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| 15 |
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| 16 | Here's the setup:
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| 17 |
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| 18 | A general degree two rational map is (generically) conjugate in at
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| 19 | most six ways to a map of the form
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| 20 |
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| 21 | f_ab = z(az^2 + bz + a) = 1/ (a (z+1/z) + b), a nonzero.
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| 22 |
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| 23 | as you can easily check using algebra. The critical points are at
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| 24 | +/- 1 and infinity maps to zero, which is fixed.
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| 25 | Such a map has three fixed points whose multipliers are, say, m1, m2, m3.
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| 26 | Let s1=m1+m2+m3 and s2=m1*m2 + m1*m3 + m2*m3 be the first two
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| 27 | elementary symmetric functions of these multipliers. From work of
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| 28 | Milnor (generalized by Silverman) one knows that s1 and s2 give canonical
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| 29 | coordinates for the moduli space of degree two rational maps, i.e. the space
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| 30 | of rational maps of degree two modulo moebius conjugation is biholomorphic
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| 31 | to C^2 with these canonical coordinates s1 and s2.
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| 32 |
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| 33 | One can easily check that
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| 34 |
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| 35 | a*s1-4*a^2-1+b^2+2*a = 0
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| 36 | and (E1)
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| 37 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2 = 0.
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| 38 |
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| 39 | Now, let s denote the ith elementary symmetric function of the multipliers
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| 40 | of the points of period n. From work of Silverman,
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| 41 |
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| 42 | s is a polynomial with integer coefficients in s1 and s2.
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| 43 |
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| 44 | PROBLEM: Calculate s in terms of i and n.
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| 45 |
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| 46 | Even for n=3 and i=1 this is not easy to do using "brute force"
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| 47 | symbolic computation. But somehow there ought to be a way to do
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| 48 | this using this approach.
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| 49 |
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| 50 | Case n=3 and i=1:
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| 51 | -----------------
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| 52 |
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| 53 | A degree two rational map has two three-cycles, generically.
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| 54 | Let s = s(3,1) denote the sum of the multipliers of these two three-cycles.
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| 55 | We want to find s in terms of s1 and s2.
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| 56 |
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| 57 | Let
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| 58 | z00 -> z10 -> z20 -> z00
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| 59 | and (E2)
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| 60 | z01 -> z11 -> z21 -> z01
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| 61 |
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| 62 | denote the points in the two three-cycles (the arrows denote application
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| 63 | of f_ab). Then
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| 64 |
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| 65 | s = f'(z00)*f'(z10)*f'(z20)+f'(z01)*f'(z11)*f'(z21) (E3)
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| 66 |
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| 67 | and s is the sum of these two quantities and depends only on s1 and s2.
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| 68 |
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| 69 | The conditions in (E2) and (E3) can be translated into polynomial
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| 70 | equations. Adding in the equations (E1) gives us a system of poly.
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| 71 | equations. Here are the details, in a maple-readable form:
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| 72 |
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| 73 | variables:=[z21, z11, z01, z20, z10, z00, a, b, s1, s2, s];
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| 74 | equations:=
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| 75 | [a*s1-4*a^2-1+b^2+2*a,
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| 76 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2,
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| 77 | z00-z10*(a*z00^2+b*z00+a), z10-z20*(a*z10^2+b*z10+a),
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| 78 | z20-z00*(a*z20^2+b*z20+a), z01-z11*(a*z01^2+b*z01+a),
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| 79 | z11-z21*(a*z11^2+b*z11+a), z21-z01*(a*z21^2+b*z21+a),
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| 80 | s-a^3*(1-z00^2)*(1-z10^2)*(1-z20^2)-a^3*(1-z01^2)*(1-z11^2)*(1-z21^2)];
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| 81 |
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| 82 | There are, however, some degeneracy constraints: we may not allow the
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| 83 | coefficient a to be zero. Also, there will be solutions to the equations
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| 84 | where e.g. z00=z01, z10=z11, z20=z21 which we must discard. So we have
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| 85 | the following quantities which must be nonzero:
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| 86 |
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| 87 | (D1) a
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| 88 |
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| 89 | (D2) z00-z10, z00-z20, z10-z20,
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| 90 | z01-z11, z01-z21, z11-z21,
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| 91 |
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| 92 | (where points in the same cycle collide)
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| 93 | and
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| 94 |
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| 95 | (D3) z00-z01, z00-z11, z00-z21,
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| 96 | z10-z01, z10-z11, z10-z21,
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| 97 | z20-z01, z20-z11, z20-z21
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| 98 |
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| 99 | (where points in different cycles collide).
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| 100 |
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| 101 | But in fact one can reduce this set of 1 + 6 + 9 = 16
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| 102 | degeneracy conditions somewhat using the fact that the
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| 103 | zij's are period of the prime period 3 to:
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| 104 |
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| 105 | degs:=[a, z00-z10, z01-z11, z00-z01, z00-z11, z00-z21];
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| 106 |
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| 107 | which is a great improvement.
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| 108 |
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| 109 | Thus our problem is:
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| 110 |
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| 111 | Solve "equations" in "variables" subject to the
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| 112 | constraint that "degs" are nonzero, and eliminate
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| 113 | all the variables save s,s1,s2.
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| 114 |
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| 115 | An alternative approach, which you mentioned, is to first
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| 116 | omit consideration of the degeneracy equations.
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| 117 |
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| 118 | I'm curious to see how you would approach this problem
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| 119 | using what you called parameters. I'd be grateful for
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| 120 | any input you could give.
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| 121 |
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| 122 | If this one looks too easy, I can send you the data for
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| 123 | higher periods. For e.g. n=4 there are 3 four-cycles and the
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| 124 | degeneracies which can occur are more complicated.
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| 125 |
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| 126 | Another useful thing to have would be the equations in a,b, and
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| 127 | the zij's defining the locus where the values of the zij's represent
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| 128 | precisely the locations of the points in the two three-cycles
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| 129 | (i.e. calculate a basis for the ideal corresponding to the variety
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| 130 | defined by (E2) subject to the constraint that the degeneracies (D1)-(D3)
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| 131 | are nonzero. Even this looks hard.
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| 132 |
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| 133 | If you're interested I can send some samples of the equations which
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| 134 | arise when trying to find postcritically finite maps.
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| 135 |
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| 136 | I enjoyed your talk in Atlanta, and thank you for organizing a great
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| 137 | session and allowing me the opportunity to speak.
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| 138 |
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| 139 | Kevin M. Pilgrim
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| 140 | Dept. of Mathematics
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| 141 | B-35 White Hall
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| 142 | Cornell University
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| 143 | Ithaca, NY 14853-7901
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| 144 | (607) 255-5380 w
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| 145 | (607) 256-1634 h, before 9PM
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| 146 | pilgrim@math.cornell.edu
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| 147 |
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| 148 | |#
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| 149 |
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| 150 |
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| 151 | (setf main-variables '(s s1 s2))
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| 152 | (setf parameters '(a b z00 z10 z20 z01 z11 z21))
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| 153 | (setf variables (append main-variables parameters))
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| 154 |
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| 155 | (setf equations
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| 156 | "[
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| 157 | z00-z10*(a*z00^2+b*z00+a),
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| 158 | z10-z20*(a*z10^2+b*z10+a),
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| 159 | z20-z00*(a*z20^2+b*z20+a),
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| 160 | z01-z11*(a*z01^2+b*z01+a),
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| 161 | z11-z21*(a*z11^2+b*z11+a),
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| 162 | z21-z01*(a*z21^2+b*z21+a),
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| 163 | a*s1-4*a^2-1+b^2+2*a,
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| 164 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2,
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| 165 | s-a^3*(1-z00^2)*(1-z10^2)*(1-z20^2)-a^3*(1-z01^2)*(1-z11^2)*(1-z21^2)
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| 166 | ]"
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| 167 | )
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| 168 |
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| 169 | (setf parametric-equations "[
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| 170 | a*s1-4*a^2-1+b^2+2*a,
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| 171 | a^2*s2-4*a^4+4*a^3+2*a+b^2+b^2*a^2-5*a^2,
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| 172 | s-a^3*(1-z00^2)*(1-z10^2)*(1-z20^2)-a^3*(1-z01^2)*(1-z11^2)*(1-z21^2)
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| 173 | ]")
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| 174 |
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| 175 | (setf green-list "[
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| 176 | z00-z10*(a*z00^2+b*z00+a),
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| 177 | z10-z20*(a*z10^2+b*z10+a),
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| 178 | z20-z00*(a*z20^2+b*z20+a),
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| 179 | z01-z11*(a*z01^2+b*z01+a),
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| 180 | z11-z21*(a*z11^2+b*z11+a),
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| 181 | z21-z01*(a*z21^2+b*z21+a)
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| 182 | ]")
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| 183 |
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| 184 | (setf red-list "[a,
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| 185 | z00-z10, z00-z20, z10-z20,
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| 186 | z01-z11, z01-z21, z11-z21,
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| 187 | z00-z01, z00-z11, z00-z21,
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| 188 | z10-z01, z10-z11, z10-z21,
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| 189 | z20-z01, z20-z11, z20-z21,
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| 190 | z00-z10,
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| 191 | z01-z11,
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| 192 | z00-z01,
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| 193 | z00-z11,
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| 194 | z00-z21]")
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| 195 |
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| 196 | (setf order #'grevlex>)
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| 197 |
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| 198 |
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| 199 | #|
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| 200 | (string-grobner
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| 201 | equations
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| 202 | variables
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| 203 | :order order)
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| 204 | |#
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| 205 |
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| 206 | #|
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| 207 | (string-ideal-polysaturation-1
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| 208 | equations
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| 209 | red-list
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| 210 | variables
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| 211 | :order order
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| 212 | )
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| 213 | |#
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| 214 |
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| 215 | #|
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| 216 | (string-grobner-system
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| 217 | parametric-equations
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| 218 | main-variables
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| 219 | parameters
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| 220 | :cover (list (list green-list red-list))
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| 221 | :main-order #'grevlex>
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| 222 | :parameter-order #'grevlex>)
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| 223 | |#
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| 224 |
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| 225 | #|
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| 226 | (string-grobner
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| 227 | green-list
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| 228 | parameters
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| 229 | :order #'grevlex>)
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| 230 | |#
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| 231 |
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| 232 | (string-ideal-polysaturation-1
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| 233 | green-list
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| 234 | red-list
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| 235 | parameters
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| 236 | :order order
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| 237 | )
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