[380a17b] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[54f69bc] | 2 | version="version divisors.lib 4.2.0.1 Jan_2021 "; // $Id$ |
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[1e1ec4] | 3 | category = "Commutative Algebra"; |
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| 4 | info=" |
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| 5 | LIBRARY: divisors.lib Divisors and P-Divisors |
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| 6 | |
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| 7 | AUTHORS: Janko Boehm boehm@mathematik.uni-kl.de |
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| 8 | @* Lars Kastner kastner@math.fu-berlin.de |
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| 9 | @* Benjamin Lorenz blorenz@math.uni-frankfurt.de |
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| 10 | @* Hans Schoenemann hannes@mathematik.uni-kl.de |
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| 11 | @* Yue Ren ren@mathematik.uni-kl.de |
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| 12 | |
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| 13 | OVERVIEW: |
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| 14 | |
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| 15 | We implement a class divisor on an algebraic variety and methods |
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| 16 | for computing with them. Divisors are represented by tuples of ideals |
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| 17 | defining the positive and the negative part. In particular, we implement the group |
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| 18 | structure on divisors, computing global sections and testing linear |
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| 19 | equivalence. |
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| 20 | |
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| 21 | In addition to this we provide a class formaldivisor which implements |
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| 22 | integer formal sums of divisors (not necessarily prime). A formal divisor |
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| 23 | can be evaluated to a divisor, and a divisor can be decomposed into |
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| 24 | a formal sum. |
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| 25 | |
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| 26 | Finally we provide a class pdivisor which implements polyhedral formal sums of |
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| 27 | divisors (P-divisors) where the coefficients are assumed to be polyhedra with fixed tail cone. |
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| 28 | There is a function to evaluate a P-divisor on a vector in the dual of the tail cone. The |
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| 29 | result will be a formal divisor. |
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| 30 | |
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| 31 | |
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| 32 | REFERENCES: |
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| 33 | |
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| 34 | For the class divisor we closely follow Macaulay2's tutorial on divisors. |
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| 35 | |
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| 36 | |
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| 37 | PROCEDURES: |
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| 38 | |
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| 39 | makeDivisor(ideal,ideal) create a divisor |
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| 40 | divisorplus(divisor,divisor) add two divisors |
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[20f2239] | 41 | multdivisor(int,divisor) multiply a divisor by an integer |
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[1e1ec4] | 42 | negativedivisor(divisor) compute the negative of the divisor |
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| 43 | normalForm(divisor) normal form of a divisor |
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| 44 | isEqualDivisor(divisor,divisor) test whether two divisors are equal |
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| 45 | globalSections(divisor) compute the global sections of a divisor |
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| 46 | degreeDivisor(divisor) degree of a divisor |
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| 47 | linearlyEquivalent(divisor,divisor) test whether two divisors a linearly equivalent |
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| 48 | effective(divisor) compute an effective divisor |
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| 49 | linearly equivalent to a divisor |
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| 50 | |
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| 51 | makeFormalDivisor(list) make a formal integer sum of divisors |
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[20f2239] | 52 | evaluateFormalDivisor(formaldivisor) evaluate a formal sum of divisors to a divisor |
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[1e1ec4] | 53 | formaldivisorplus(formaldivisor,formaldivisor) add two formal divisors |
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| 54 | negativeformaldivisor(formaldivisor) compute the negative of the formal divisor |
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[20f2239] | 55 | multformaldivisor(int,formaldivisor) multiply a formal divisor by an integer |
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[1e1ec4] | 56 | degreeFormalDivisor(formaldivisor) degree of a formal divisor |
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| 57 | |
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| 58 | makePDivisor(List) make a formal polyhedral sum of divisors |
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| 59 | |
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| 60 | |
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| 61 | "; |
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| 62 | |
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| 63 | |
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| 64 | //////////////////////////////////////////////////////////////////////////////// |
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| 65 | |
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| 66 | LIB "primdec.lib"; |
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| 67 | |
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[54f69bc] | 68 | static proc mod_init() |
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[1e1ec4] | 69 | { |
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[e861ae] | 70 | LIB"customstd.so"; |
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[1243bc] | 71 | LIB"gfanlib.so"; |
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[1e1ec4] | 72 | newstruct("divisor","ideal den,ideal num"); |
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| 73 | newstruct("formaldivisor","list summands"); |
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| 74 | newstruct("pdivisor","list summands, cone tail"); |
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| 75 | |
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| 76 | system("install","divisor","print",divisor_print,1); |
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| 77 | system("install","divisor","+",divisorplus,2); |
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| 78 | system("install","divisor","*",proxymultdivisor,2); |
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| 79 | system("install","formaldivisor","print",formaldivisor_print,1); |
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| 80 | system("install","formaldivisor","+",formaldivisorplus,2); |
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| 81 | system("install","formaldivisor","*",proxymultformaldivisor,2); |
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| 82 | } |
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| 83 | |
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| 84 | |
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| 85 | |
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| 86 | proc divisor_print(divisor D) |
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| 87 | "USAGE: divisor_print(D); D; D = divisor. @* |
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| 88 | ASSUME: D is a divisor. |
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| 89 | RETURN: Will print D. |
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| 90 | KEYWORDS: divisors |
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| 91 | EXAMPLE: example divisor_print; shows an example |
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| 92 | " |
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| 93 | { |
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| 94 | "("+string(D.num)+") - ("+string(D.den)+")"; |
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| 95 | } |
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| 96 | example |
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| 97 | { "EXAMPLE:"; |
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| 98 | ring r=31991,(x,y,z),dp; |
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| 99 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 100 | qring Q = std(I); |
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| 101 | divisor P = makeDivisor(ideal(x,z),ideal(1)); |
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| 102 | P; |
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| 103 | } |
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| 104 | |
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| 105 | |
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| 106 | proc formaldivisor_print(formaldivisor fD) |
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| 107 | "USAGE: formaldivisor_print(D); D; D = formaldivisor. @* |
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| 108 | ASSUME: fD is a formaldivisor. |
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| 109 | RETURN: Will print fD. |
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| 110 | KEYWORDS: divisors |
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| 111 | EXAMPLE: example formaldivisor_print; shows an example |
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| 112 | " |
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| 113 | { |
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| 114 | int i; string s; list L=fD.summands; |
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| 115 | list cd; int c; divisor d; |
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| 116 | string linebreak = |
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| 117 | " |
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| 118 | "; |
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| 119 | for (i=1; i<=size(L); i++) |
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| 120 | { |
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| 121 | cd=L[i]; c=cd[1]; d=cd[2]; |
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| 122 | if (i>1 && c>=0) { s = s + "+"; } |
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| 123 | s = s + string(c)+"*( ("+string(d.num)+") - ("+string(d.den)+") )"; |
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| 124 | if (i!=size(L)) { s = s + linebreak; } |
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| 125 | } |
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| 126 | s; |
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| 127 | } |
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| 128 | example |
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| 129 | { "EXAMPLE:"; |
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| 130 | ring r=31991,(x,y,z),dp; |
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| 131 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 132 | qring Q = std(I); |
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| 133 | divisor P = makeDivisor(ideal(x,z),ideal(1)); |
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| 134 | P; |
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| 135 | } |
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| 136 | |
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| 137 | |
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| 138 | //////////////////////////////////////////////////////////////////////////////// |
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| 139 | // divisors as numerator and denomiator ideals |
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| 140 | |
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| 141 | proc makeDivisor(ideal I, ideal J) |
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| 142 | "USAGE: makeDivisor(I ,J); I = ideal, J = ideal. @* |
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| 143 | ASSUME: I and J are ideals in a qring Q of a smooth irreducible variety X |
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| 144 | such that any ideal in Q satisfies the S2 condition. |
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| 145 | RETURN: a divisor on X |
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| 146 | THEORY: The procedure will eliminate all components which are not of codimension 1. |
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| 147 | The S2 condition requires that every proper nonzero principal ideal |
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| 148 | has pure codimension 1. |
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| 149 | KEYWORDS: divisors |
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| 150 | EXAMPLE: example makeDivisor; shows an example |
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| 151 | " |
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| 152 | { |
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| 153 | divisor C; |
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| 154 | C.num = purify1(I); |
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| 155 | C.den = purify1(J); |
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| 156 | return(C); |
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| 157 | } |
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| 158 | |
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| 159 | example |
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| 160 | { "EXAMPLE:"; |
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| 161 | ring r=31991,(x,y,z),dp; |
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| 162 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 163 | qring Q = std(I); |
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| 164 | divisor P = makeDivisor(ideal(x,z),ideal(1)); |
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| 165 | } |
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| 166 | |
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| 167 | |
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| 168 | proc divisorplus(divisor A, divisor B) |
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| 169 | "USAGE: divisorplus(A ,B); A + B; A = divisor, B = divisor. @* |
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| 170 | ASSUME: A and B are divisors on X. |
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| 171 | RETURN: a divisor on X |
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| 172 | THEORY: The procedure will compute the product of the numerator |
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| 173 | and denominator ideals, respectively. |
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| 174 | KEYWORDS: divisors |
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| 175 | EXAMPLE: example divisorplus; shows an example |
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| 176 | " |
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| 177 | { |
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| 178 | return(makeDivisor(interred(A.num*B.num),interred(A.den*B.den))); |
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| 179 | } |
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| 180 | example |
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| 181 | { "EXAMPLE:"; |
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| 182 | ring r=31991,(x,y,z),dp; |
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| 183 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 184 | qring Q = std(I); |
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| 185 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 186 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 187 | A+B; |
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| 188 | } |
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| 189 | |
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| 190 | |
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| 191 | proc multdivisor(int n, divisor A) |
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| 192 | "USAGE: multdivisor(n ,A); A*n; n = integer, A = divisor.@* |
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| 193 | ASSUME: n is an integer and A is a divisor on X. |
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| 194 | RETURN: a divisor on X |
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| 195 | THEORY: The procedure will compute the n-th power of the numerator |
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| 196 | and denominator ideals, respectively. |
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| 197 | KEYWORDS: divisors |
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| 198 | EXAMPLE: example multdivisor; shows an example |
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| 199 | " |
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| 200 | { |
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| 201 | if (n<0) {return(multdivisor(-n,negativedivisor(A)));} |
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| 202 | return(makeDivisor(interred((A.num)^n),interred((A.den)^n))); |
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| 203 | } |
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| 204 | example |
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| 205 | { "EXAMPLE:"; |
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| 206 | ring r=31991,(x,y,z),dp; |
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| 207 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 208 | qring Q = std(I); |
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| 209 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 210 | A; |
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| 211 | divisor D = multdivisor(4,A); |
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| 212 | D; |
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| 213 | A*4; |
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| 214 | } |
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| 215 | |
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| 216 | |
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| 217 | /*** |
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| 218 | * For operator overloading, which needs a procedure which takes a divisor first |
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| 219 | * and integer second. |
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| 220 | **/ |
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| 221 | proc proxymultdivisor(divisor A, int n) |
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| 222 | { |
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| 223 | if (n<0) {return(multdivisor(-n,negativedivisor(A)));} |
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| 224 | return(makeDivisor(interred((A.num)^n),interred((A.den)^n))); |
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| 225 | } |
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| 226 | |
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| 227 | |
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| 228 | proc negativedivisor(divisor A) |
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| 229 | "USAGE: negativedivisor(A); A*(-1); A = divisor.@* |
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| 230 | ASSUME: A is a divisor on X. |
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| 231 | RETURN: a divisor on X |
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| 232 | THEORY: The procedure will interchange the numerator and denominator ideals. |
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| 233 | KEYWORDS: divisors |
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| 234 | EXAMPLE: example negativedivisor; shows an example |
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| 235 | " |
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| 236 | { |
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| 237 | divisor B; |
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| 238 | B.num=A.den; |
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| 239 | B.den=A.num; |
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| 240 | return(B); |
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| 241 | } |
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| 242 | example |
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| 243 | { "EXAMPLE:"; |
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| 244 | ring r=31991,(x,y,z),dp; |
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| 245 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 246 | qring Q = std(I); |
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| 247 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 248 | A; |
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| 249 | divisor D = negativedivisor(A); |
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| 250 | D; |
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| 251 | } |
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| 252 | |
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| 253 | |
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| 254 | proc normalForm(divisor A) |
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| 255 | "USAGE: normalForm(A); A = divisor.@* |
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| 256 | ASSUME: A is a divisor on X. |
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| 257 | RETURN: different representative of the same divisor on X |
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| 258 | THEORY: The procedure will cancel common components of numerator and denominator. |
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| 259 | KEYWORDS: divisors |
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| 260 | EXAMPLE: example normalForm; shows an example |
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| 261 | " |
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| 262 | { |
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| 263 | divisor B; |
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| 264 | B.num=quotient(A.num,A.den); |
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| 265 | B.den=quotient(A.den,A.num); |
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| 266 | return(B); |
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| 267 | } |
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| 268 | example |
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| 269 | { "EXAMPLE:"; |
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| 270 | ring r=31991,(x,y,z),dp; |
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| 271 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 272 | qring Q = std(I); |
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| 273 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 274 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 275 | divisor D = (A+B)+multdivisor(-1,B); |
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| 276 | D; |
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| 277 | normalForm(D); |
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| 278 | } |
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| 279 | |
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| 280 | |
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| 281 | |
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| 282 | |
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| 283 | static proc isEqualIdeal(ideal A,ideal B){ |
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| 284 | return((size(NF(A,std(B)))==0) && (size(NF(B,std(A)))==0)); |
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| 285 | } |
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| 286 | |
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| 287 | |
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| 288 | proc isEqualDivisor(divisor A,divisor B) |
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| 289 | "USAGE: isEqualDivisor(A,B); A = divisor, B = divisor.@* |
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| 290 | ASSUME: A and B are divisors on X. |
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| 291 | RETURN: int 0 or 1, checks equality of A and B. |
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| 292 | THEORY: The procedure will compute the normal forms of A and B and compare. |
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| 293 | KEYWORDS: divisors |
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| 294 | EXAMPLE: example isEqualDivisor; shows an example |
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| 295 | " |
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| 296 | { |
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| 297 | A=normalForm(A); |
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| 298 | B=normalForm(B); |
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| 299 | return((isEqualIdeal(A.num,B.num)) && (isEqualIdeal(A.den,B.den))); |
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| 300 | } |
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| 301 | example |
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| 302 | { "EXAMPLE:"; |
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| 303 | ring r=31991,(x,y,z),dp; |
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| 304 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 305 | qring Q = std(I); |
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| 306 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 307 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 308 | divisor D = (A+B)+multdivisor(-1,B); |
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| 309 | isEqualDivisor(A,D); |
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| 310 | } |
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| 311 | |
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| 312 | |
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| 313 | static proc purify1(ideal I) |
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| 314 | { |
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| 315 | I = simplify(I,2); |
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| 316 | if (I[1]==0){ERROR("expected a non-zero ideal");} |
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| 317 | ideal f = I[1]; |
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| 318 | return(minbase(quotient(f,quotient(f,I)))); |
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| 319 | } |
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| 320 | |
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| 321 | |
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| 322 | static proc basis(ideal I,int d) |
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| 323 | { |
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| 324 | I=simplify(jet(intersect(I,maxideal(d)),d),2); |
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| 325 | return(I)} |
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| 326 | |
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| 327 | //basis(ideal(x,y^3),2); |
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| 328 | |
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| 329 | |
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| 330 | proc globalSections(divisor D) |
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| 331 | "USAGE: globalSections(A); A = divisor.@* |
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| 332 | ASSUME: A is a divisor on X. |
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| 333 | RETURN: a list with a basis of the space of global sections of D. |
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[389c6d] | 334 | THEORY: We assume that the qring of X satisfies the S2-condition and that X is smooth. We compute sat((f*J) : I) /f |
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[1e1ec4] | 335 | where D = (I)-(J). |
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| 336 | KEYWORDS: divisors |
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| 337 | EXAMPLE: example globalSections; shows an example |
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| 338 | " |
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| 339 | { |
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| 340 | poly f =(simplify(D.num,2))[1]; |
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| 341 | ideal LD = basis(purify1(quotient(f*D.den,D.num)),deg(f)); |
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| 342 | list L = simplify(LD,2),f; |
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| 343 | return(L); |
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| 344 | } |
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| 345 | example |
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| 346 | { "EXAMPLE:"; |
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| 347 | ring r=31991,(x,y,z),dp; |
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| 348 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 349 | qring Q = std(I); |
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| 350 | divisor P = makeDivisor(ideal(x,z),ideal(1)); |
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| 351 | divisor D = multdivisor(4,P); |
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| 352 | globalSections(D); |
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| 353 | } |
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| 354 | |
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| 355 | |
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| 356 | static proc sectionIdeal(poly f, poly g, divisor D){ |
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| 357 | return(purify1(quotient(quotient(f*D.num,g), D.den))); |
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| 358 | } |
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| 359 | |
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| 360 | proc degreeDivisor(divisor A) |
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| 361 | "USAGE: degreeDivisor(A); A = divisor.@* |
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| 362 | ASSUME: A is a divisor on X. |
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| 363 | RETURN: The degree of A. |
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| 364 | THEORY: We compute difference of the degrees of the numerator and denominator ideals. |
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| 365 | KEYWORDS: divisors |
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| 366 | EXAMPLE: example degreeDivisor; shows an example |
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| 367 | " |
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| 368 | { |
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[389c6d] | 369 | return( mult(std(A.num))-mult(std(A.den))); |
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[1e1ec4] | 370 | } |
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| 371 | example |
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| 372 | { "EXAMPLE:"; |
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| 373 | ring r=31991,(x,y,z),dp; |
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| 374 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 375 | qring Q = std(I); |
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| 376 | divisor P = makeDivisor(ideal(x,z),ideal(1)); |
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| 377 | degreeDivisor(P); |
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| 378 | } |
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| 379 | |
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| 380 | proc linearlyEquivalent(divisor A, divisor B) |
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| 381 | "USAGE: linearlyEquivalent(A,B); A = divisor, B = divisor.@* |
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| 382 | ASSUME: A and B are divisors on X. |
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| 383 | RETURN: list if A and B a linearly equivalent and int 0 otherwise. |
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| 384 | THEORY: Checks whether A-B is principle. If yes, returns a list L=(f,g) where |
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| 385 | A - B = (f/g). |
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| 386 | KEYWORDS: divisors |
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| 387 | EXAMPLE: example linearlyEquivalent; shows an example |
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| 388 | " |
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| 389 | { |
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| 390 | divisor F = normalForm(divisorplus(A,negativedivisor(B))); |
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| 391 | list LB = globalSections(F); |
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| 392 | if (size(LB[1])!=1) {return(0);} |
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| 393 | ideal V = sectionIdeal(LB[1][1,1],LB[2],F); |
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| 394 | if (isEqualIdeal(V,ideal(1))==1) {return(list(LB[1][1,1],LB[2]));} |
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| 395 | return(0); |
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| 396 | } |
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| 397 | example |
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| 398 | { "EXAMPLE:"; |
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| 399 | ring r=31991,(x,y,z),dp; |
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| 400 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 401 | qring Q = std(I); |
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| 402 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 403 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 404 | linearlyEquivalent(A,B); |
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| 405 | linearlyEquivalent(multdivisor(2,A),multdivisor(2,B)); |
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| 406 | } |
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| 407 | |
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| 408 | |
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| 409 | proc effective(divisor A) |
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| 410 | "USAGE: effective(A); A = divisor.@* |
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| 411 | ASSUME: A is a divisor on X which is linearly equivalent to an effective divisor. |
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| 412 | RETURN: divisor on X. |
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| 413 | THEORY: We compute an effective divisor linearly equivalent to A. |
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| 414 | KEYWORDS: divisors |
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| 415 | EXAMPLE: example effective; shows an example |
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| 416 | " |
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| 417 | { |
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| 418 | list LB = globalSections(A); |
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| 419 | if (size(LB[1])==0) {ERROR("the divisor is not linearly equivalent to an effective divisor");} |
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| 420 | ideal V = sectionIdeal(LB[1][1,1],LB[2],A); |
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| 421 | return(makeDivisor(V,ideal(1))); |
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| 422 | } |
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| 423 | example |
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| 424 | { "EXAMPLE:"; |
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| 425 | ring r=31991,(x,y,z),dp; |
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| 426 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 427 | qring Q = std(I); |
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| 428 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 429 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 430 | divisor D = divisorplus(multdivisor(2,B),negativedivisor(A)); |
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| 431 | effective(D); |
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| 432 | } |
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| 433 | |
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| 434 | |
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| 435 | |
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| 436 | //////////////////////////////////////////////////////////////////////////////// |
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| 437 | // formal sums of divisors |
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| 438 | |
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| 439 | proc makeFormalDivisor(list L) |
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| 440 | "USAGE: makeFormalDivisor(L); L = list.@* |
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| 441 | ASSUME: L is a list of tuples of an integer and a divisor. |
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| 442 | RETURN: a formal divisor on X |
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| 443 | THEORY: Represents an integer formal sum of divisors. |
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| 444 | KEYWORDS: divisors |
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| 445 | EXAMPLE: example makeFormalDivisor; shows an example |
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| 446 | " |
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| 447 | { |
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| 448 | formaldivisor C; |
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| 449 | C.summands = L; |
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| 450 | return(C); |
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| 451 | } |
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| 452 | example |
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| 453 | { "EXAMPLE:"; |
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| 454 | ring r=31991,(x,y,z),dp; |
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| 455 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 456 | qring Q = std(I); |
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| 457 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 458 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 459 | makeFormalDivisor(list(list(-5,A),list(2,B))); |
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| 460 | } |
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| 461 | |
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| 462 | |
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| 463 | proc evaluateFormalDivisor(formaldivisor D) |
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| 464 | "USAGE: evaluateFormalDivisor(D); D = formal divisor.@* |
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| 465 | ASSUME: D is a formal divisor on X. |
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| 466 | RETURN: a divisor on X |
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| 467 | THEORY: Will evaluate the formal sum. |
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| 468 | KEYWORDS: divisors |
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| 469 | EXAMPLE: example evaluateFormalDivisor; shows an example |
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| 470 | " |
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| 471 | { |
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| 472 | list L = D.summands; |
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| 473 | if (size(L)==0) {return(makeDivisor(ideal(1),ideal(1)));} |
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| 474 | int i; |
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| 475 | divisor E = multdivisor(L[1][1],L[1][2]); |
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| 476 | for ( i=2; i <= size(L); i++ ) |
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| 477 | { |
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| 478 | E = divisorplus(E, multdivisor(L[i][1],L[i][2])); |
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| 479 | } |
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| 480 | return(E); |
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| 481 | } |
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| 482 | example |
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| 483 | { "EXAMPLE:"; |
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| 484 | ring r=31991,(x,y,z),dp; |
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| 485 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 486 | qring Q = std(I); |
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| 487 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 488 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 489 | formaldivisor fE= makeFormalDivisor(list(list(-5,A),list(2,B))); |
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| 490 | evaluateFormalDivisor(fE); |
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| 491 | } |
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| 492 | |
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| 493 | |
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| 494 | |
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| 495 | static proc position(divisor I,list L){ |
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| 496 | int i; |
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| 497 | for (i=1; i <=size(L); i++){ |
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| 498 | if (isEqualDivisor(I,L[i][2])==1) {return(i);} |
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| 499 | } |
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| 500 | return(0);} |
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| 501 | |
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| 502 | |
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| 503 | proc formaldivisorplus(formaldivisor A, formaldivisor B) |
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| 504 | "USAGE: formaldivisorplus(A ,B); A + B; A = formaldivisor, B = formaldivisor. @* |
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| 505 | ASSUME: A and B are formal divisors on X. |
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| 506 | RETURN: a formal divisor on X |
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| 507 | THEORY: The procedure will add the formal sums. |
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| 508 | KEYWORDS: divisors |
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| 509 | EXAMPLE: example formaldivisorplus; shows an example |
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| 510 | " |
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| 511 | { |
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| 512 | formaldivisor C; |
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| 513 | int i,p; |
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| 514 | int j=1; |
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| 515 | list L; |
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| 516 | list LA=A.summands; |
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| 517 | list LB=B.summands; |
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| 518 | for (i=1; i<=size(LA);i++){ |
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| 519 | p=position(LA[i][2],L); |
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| 520 | if (p==0) { |
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| 521 | L[j]=list(); |
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| 522 | L[j][2]=LA[i][2]; |
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| 523 | L[j][1]=LA[i][1]; |
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| 524 | j=j+1; |
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| 525 | } else { |
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| 526 | L[p][1]=L[p][1]+LA[i][1]; |
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| 527 | }; |
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| 528 | } |
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| 529 | for (i=1; i<=size(LB);i++){ |
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| 530 | p=position(LB[i][2],L); |
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| 531 | if (p==0) { |
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| 532 | L[j]=list(); |
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| 533 | L[j][2]=LB[i][2]; |
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| 534 | L[j][1]=LB[i][1]; |
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| 535 | j=j+1; |
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| 536 | } else { |
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| 537 | L[p][1]=L[p][1]+LB[i][1]; |
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| 538 | }; |
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| 539 | } |
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| 540 | //C.summands = (A.summands)+(B.summands); |
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| 541 | return(L); |
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| 542 | } |
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| 543 | example |
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| 544 | { "EXAMPLE:"; |
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| 545 | ring r=31991,(x,y,z),dp; |
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| 546 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 547 | qring Q = std(I); |
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| 548 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 549 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 550 | divisor C = makeDivisor(ideal(x-z,y),ideal(1)); |
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| 551 | formaldivisor fE= makeFormalDivisor(list(list(-5,A),list(2,B))); |
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| 552 | formaldivisor fE2= makeFormalDivisor(list(list(-5,A),list(2,C))); |
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| 553 | formaldivisorplus(fE,fE2); |
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| 554 | } |
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| 555 | |
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| 556 | |
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| 557 | proc degreeFormalDivisor(formaldivisor A) |
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| 558 | "USAGE: degreeFormalDivisor(A); A = formaldivisor.@* |
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| 559 | ASSUME: A is a formaldivisor on X. |
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| 560 | RETURN: The degree of A. |
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| 561 | THEORY: We compute degrees of the summands and return the weighted sum. |
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| 562 | KEYWORDS: divisors |
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| 563 | EXAMPLE: example degreeFormalDivisor; shows an example |
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| 564 | " |
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| 565 | { |
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| 566 | int i,s; |
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| 567 | list L = A.summands; |
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| 568 | for (i=1;i<=size(L);i++){ |
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| 569 | s=s+L[i][1]*degreeDivisor(L[i][2]); |
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| 570 | } |
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| 571 | return(s); |
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| 572 | } |
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| 573 | example |
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| 574 | { "EXAMPLE:"; |
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| 575 | ring r=31991,(x,y,z),dp; |
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| 576 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 577 | qring Q = std(I); |
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| 578 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 579 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 580 | formaldivisor fE= makeFormalDivisor(list(list(-5,A),list(2,B))); |
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| 581 | degreeFormalDivisor(fE); |
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| 582 | } |
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| 583 | |
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| 584 | |
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| 585 | proc multformaldivisor(int n,formaldivisor A) |
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| 586 | "USAGE: multformaldivisor(n ,A); A*n; n = integer, A = formaldivisor.@* |
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| 587 | ASSUME: n is an integer and A is a formal divisor on X. |
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| 588 | RETURN: a formal divisor on X |
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| 589 | THEORY: The procedure will multiply the formal sum with n. |
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| 590 | KEYWORDS: divisors |
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| 591 | EXAMPLE: example multformaldivisor; shows an example |
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| 592 | " |
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| 593 | { |
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| 594 | formaldivisor B; |
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| 595 | list L=A.summands; |
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| 596 | int i; |
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| 597 | for (i=1;i<=size(L);i++){ |
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| 598 | L[i][1]=n*L[i][1]; |
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| 599 | } |
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| 600 | B.summands=L; |
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| 601 | return(B); |
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| 602 | } |
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| 603 | example |
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| 604 | { "EXAMPLE:"; |
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| 605 | ring r=31991,(x,y,z),dp; |
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| 606 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 607 | qring Q = std(I); |
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| 608 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 609 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 610 | formaldivisor fE= makeFormalDivisor(list(list(-5,A),list(2,B))); |
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| 611 | fE*2; |
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| 612 | } |
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| 613 | |
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| 614 | |
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| 615 | /*** |
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| 616 | * For operator overloading, which needs a procedure which takes a divisor first |
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| 617 | * and integer second. |
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| 618 | **/ |
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| 619 | proc proxymultformaldivisor(formaldivisor A, int n) |
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| 620 | { |
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| 621 | formaldivisor B; |
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| 622 | list L=A.summands; |
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| 623 | int i; |
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| 624 | for (i=1;i<=size(L);i++){ |
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| 625 | L[i][1]=n*L[i][1]; |
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| 626 | } |
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| 627 | B.summands=L; |
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| 628 | return(B); |
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| 629 | } |
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| 630 | |
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| 631 | |
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| 632 | |
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| 633 | proc negativeformaldivisor(formaldivisor A) |
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| 634 | "USAGE: negativeformaldivisor(A); A = formaldivisor.@* |
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| 635 | ASSUME: A is a formaldivisor on X. |
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| 636 | RETURN: a formal divisor on X |
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| 637 | THEORY: The procedure will change the signs of the coefficients. |
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| 638 | KEYWORDS: divisors |
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| 639 | EXAMPLE: example negativeformaldivisor; shows an example |
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| 640 | " |
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| 641 | { |
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| 642 | formaldivisor B; |
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| 643 | list L=A.summands; |
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| 644 | int i; |
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| 645 | for (i=1;i<=size(L);i++){ |
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| 646 | L[i][1]=-L[i][1]; |
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| 647 | } |
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| 648 | B.summands=L; |
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| 649 | return(B); |
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| 650 | } |
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| 651 | example |
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| 652 | { "EXAMPLE:"; |
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| 653 | ring r=31991,(x,y,z),dp; |
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| 654 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 655 | qring Q = std(I); |
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| 656 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 657 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 658 | formaldivisor fE= makeFormalDivisor(list(list(-5,A),list(2,B))); |
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| 659 | negativeformaldivisor(fE); |
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| 660 | } |
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| 661 | |
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| 662 | |
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| 663 | static proc primDecDivisor(divisor D) |
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| 664 | "decompose a divisor into a formal divisor of primary divisors" |
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| 665 | { |
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| 666 | formaldivisor E; |
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| 667 | ideal I = D.num; |
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| 668 | ideal J = D.den; |
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| 669 | list L; |
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| 670 | int i; |
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| 671 | int j = 1; |
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| 672 | list LI = primdecGTZ(I); |
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| 673 | for (i=1;i<=size(LI);i++){ |
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| 674 | LI[i][2]; |
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| 675 | L[j]=list(1,makeDivisor(LI[i][1],ideal(1))); |
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| 676 | j=j+1; |
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| 677 | }; |
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| 678 | list LJ = primdecGTZ(J); |
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| 679 | for (i=1;i<=size(LJ);i++){ |
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| 680 | LJ[i][2]; |
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| 681 | L[j]=list(-1,makeDivisor(LJ[i][1],ideal(1))); |
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| 682 | j=j+1; |
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| 683 | }; |
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| 684 | E.summands=L; |
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| 685 | return(E);} |
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| 686 | |
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| 687 | |
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| 688 | |
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| 689 | |
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| 690 | //////////////////////////////////////////////////////////////////////////////// |
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| 691 | // P-divisors |
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| 692 | |
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| 693 | proc makePDivisor(list L) |
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| 694 | "USAGE: makePDivisor(L); L = list.@* |
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| 695 | ASSUME: L is a list of tuples of a integral polyhedron and a divisor such that |
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| 696 | all polyhedra have the same tail cone. |
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| 697 | RETURN: a pdivisor on X |
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| 698 | THEORY: Represents an polyhedral formal sum of divisors. |
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| 699 | KEYWORDS: divisors, polyhedra |
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| 700 | EXAMPLE: example makePDivisor; shows an example |
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| 701 | " |
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| 702 | { |
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| 703 | pdivisor P; |
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| 704 | list CP = decomposePolyhedron(L[1][1]); |
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| 705 | P.tail = CP[1]; |
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| 706 | list LP; |
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| 707 | LP[1]=list(CP[2],L[1][2]); |
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| 708 | int i; |
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| 709 | for (i=2; i<=size(L);i++){ |
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| 710 | CP = decomposePolyhedron(L[i][1]); |
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| 711 | if (!(CP[1]==P.tail)) {ERROR("All P-coefficients should have the same tail cone");} |
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| 712 | LP[i]=list(CP[2],L[i][2]); |
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| 713 | } |
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| 714 | P.summands = LP; |
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| 715 | return(P); |
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| 716 | } |
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| 717 | example |
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| 718 | { "EXAMPLE:"; |
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| 719 | ring r=31991,(x,y,z),dp; |
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| 720 | ideal I = y^2*z - x*(x-z)*(x+3*z); |
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| 721 | qring Q = std(I); |
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| 722 | divisor A = makeDivisor(ideal(x,z),ideal(1)); |
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| 723 | divisor B = makeDivisor(ideal(x,y),ideal(1)); |
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| 724 | intmat M[4][4]= 1,4,0,0, |
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| 725 | 1,0,3,0, |
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| 726 | 0,0,0,2, |
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| 727 | 1,1,1,1; |
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| 728 | polytope PP = polytopeViaPoints(M); |
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| 729 | makePDivisor(list(list(PP,A),list(PP,B))); |
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| 730 | } |
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| 731 | |
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| 732 | static proc decomposePolyhedron(polytope P){ |
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[8bb9a3] | 733 | bigintmat rays = vertices(P); |
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| 734 | bigintmat rays2 = rays; |
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[1e1ec4] | 735 | int i,j; |
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| 736 | for (i=1; i<=nrows(rays);i++){ |
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| 737 | if (rays[i,1]==1) { |
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| 738 | for (j=1; j<=ncols(rays);j++){ |
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| 739 | rays[i,j]=0; |
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| 740 | } |
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| 741 | } else { |
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| 742 | for (j=1; j<=ncols(rays);j++){ |
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| 743 | rays2[i,j]=0; |
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| 744 | } |
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| 745 | } |
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| 746 | } |
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| 747 | cone C = coneViaPoints(rays); |
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| 748 | polytope C2 = polytopeViaPoints(rays2); |
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| 749 | return(list(C,C2)); |
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| 750 | } |
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| 751 | |
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| 752 | |
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| 753 | |
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