[121920] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[341696] | 2 | version="$Id$"; |
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[121920] | 3 | category="Teaching"; |
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| 4 | info=" |
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[1a3911] | 5 | LIBRARY: hyperel.lib |
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[121920] | 6 | AUTHOR: Markus Hochstetter, markushochstetter@gmx.de |
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| 7 | |
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[1a3911] | 8 | NOTE: The library provides procedures for computing with divisors in the |
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| 9 | jacobian of hyperelliptic curves. In addition procedures are available |
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| 10 | for computing the rational representation of divisors and vice versa. |
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| 11 | The library is intended to be used for teaching and demonstrating |
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| 12 | purposes but not for efficient computations. |
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[121920] | 13 | |
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| 14 | |
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| 15 | |
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| 16 | |
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| 17 | PROCEDURES: |
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| 18 | ishyper(h,f) test, if y^2+h(x)y=f(x) is hyperelliptic |
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| 19 | isoncurve(P,h,f) test, if point P is on C: y^2+h(x)y=f(x) |
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| 20 | chinrestp(b,moduli) compute polynom x, s.t. x=b[i] mod moduli[i] |
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| 21 | norm(a,b,h,f) norm of a(x)-b(x)y in IF[C] |
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| 22 | multi(a,b,c,d,h,f) (a(x)-b(x)y)*(c(x)-d(x)y) in IF[C] |
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| 23 | ratrep (P,h,f) returns polynomials a,b, s.t. div(a,b)=P |
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| 24 | divisor(a,b,h,f,[]) computes divisor of a(x)-b(x)y |
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| 25 | gcddivisor(p,q) gcd of the divisors p and q |
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| 26 | semidiv(D,h,f) semireduced divisor of the pair of polys D[1], D[2] |
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| 27 | cantoradd(D,Q,h,f) adding divisors of the hyperell. curve y^2+h(x)y=f(x) |
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| 28 | cantorred(D,h,f) returns reduced divisor which is equivalent to D |
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| 29 | double(D,h,f) computes 2*D on y^2+h(x)y=f(x) |
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| 30 | cantormult(m,D,h,f) computes m*D on y^2+h(x)y=f(x) |
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| 31 | |
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| 32 | [parameters in square brackets are optional] |
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| 33 | "; |
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| 34 | /////////////////////////////////////////////////////////////////////////////// |
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| 35 | |
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| 36 | |
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| 37 | //=============== Test, if a given curve is hyperelliptic ===================== |
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| 38 | |
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| 39 | proc ishyper(poly h, poly f) |
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[1a3911] | 40 | "USAGE: ishyper(h,f); h,f=poly |
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| 41 | RETURN: 1 if y^2+h(x)y=f(x) is hyperelliptic, 0 otherwise |
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[121920] | 42 | NOTE: Tests, if y^2+h(x)y=f(x) is a hyperelliptic curve. |
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| 43 | Curve is defined over basering. Additionally shows error-messages. |
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| 44 | EXAMPLE: example ishyper; shows an example |
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| 45 | " |
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| 46 | { |
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| 47 | // constructing a copy of the basering (only variable x), |
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| 48 | // with variables x,y. |
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| 49 | def R=basering; |
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| 50 | list l= ringlist(R); |
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| 51 | list ll=l[2]; |
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| 52 | ll="x","y"; |
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| 53 | l[2]=ll; |
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| 54 | intvec v= l[3][1][2]; |
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| 55 | v=v,1; |
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| 56 | l[3][1][2]=v; |
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| 57 | def s=ring(l); |
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| 58 | setring s; |
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| 59 | |
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| 60 | // test, if y^2 + hy - f is hyperelliptic. |
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| 61 | int i=1; |
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| 62 | poly h=imap(R,h); |
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| 63 | poly f=imap(R,f); |
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| 64 | poly F=y2 + h*y - f; |
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| 65 | ideal I=F, diff(F,x) , diff(F,y); |
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| 66 | ideal J=std(I); |
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| 67 | if ( J != 1 ) |
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| 68 | { |
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| 69 | i=0; |
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| 70 | "The curve is singular!"; |
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| 71 | } |
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| 72 | if ( deg(f) mod 2 != 1 ) |
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| 73 | { |
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| 74 | i=0; |
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| 75 | "The polynomial ",f," has even degree!"; |
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| 76 | } |
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| 77 | if ( leadcoef(f) != 1 ) |
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| 78 | { |
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| 79 | i=0; |
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| 80 | "The polynomial ",f," is not monic!"; |
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| 81 | } |
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| 82 | if ( 2*deg(h) > deg(f)-1 ) |
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| 83 | { |
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| 84 | i=0; |
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| 85 | "The polynomial ",h," has degree ",deg(h),"!"; |
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| 86 | } |
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| 87 | setring(R); |
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| 88 | return(i); |
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| 89 | } |
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| 90 | example |
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| 91 | { "EXAMPLE:"; echo = 2; |
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| 92 | ring R=7,x,dp; |
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| 93 | // hyperelliptic curve y^2 + h*y = f |
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| 94 | poly h=x; |
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| 95 | poly f=x5+5x4+6x2+x+3; |
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| 96 | ishyper(h,f); |
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| 97 | } |
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| 98 | |
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| 99 | |
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| 100 | //================= Test, if a given ponit is on the curve ==================== |
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| 101 | |
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| 102 | proc isoncurve(list P, poly h, poly f) |
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[1a3911] | 103 | "USAGE: isoncurve(P,h,f); h,f=poly; P=list |
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[121920] | 104 | RETURN: 1 or 0 (if P is on curve or not) |
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[1a3911] | 105 | NOTE: Tests, if P=(P[1],P[2]) is on the hyperelliptic curve y^2+h(x)y=f(x). |
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[121920] | 106 | Curve is defined over basering. |
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| 107 | EXAMPLE: example isoncurve; shows an example |
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| 108 | " |
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| 109 | { |
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| 110 | if ( P[2]^2 + subst(h,var(1),P[1])*P[2] - subst(f,var(1),P[1]) == 0 ) |
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| 111 | { |
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| 112 | return(1); |
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| 113 | } |
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| 114 | return(0); |
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| 115 | } |
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| 116 | example |
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| 117 | { "EXAMPLE:"; echo = 2; |
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| 118 | ring R=7,x,dp; |
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| 119 | // hyperelliptic curve y^2 + h*y = f |
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| 120 | poly h=x; |
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| 121 | poly f=x5+5x4+6x2+x+3; |
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| 122 | list P=2,3; |
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| 123 | isoncurve(P,h,f); |
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| 124 | } |
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| 125 | |
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| 126 | |
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| 127 | //====================== Remainder of a polynomial division =================== |
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| 128 | |
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| 129 | proc divrem(f,g) |
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| 130 | "USAGE: divrem(f,g); f,g poly |
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| 131 | RETURN: remainder of the division f/g |
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| 132 | NOTE: Computes R, s.t. f=a*g + R, and deg(R) < deg(g) |
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| 133 | EXAMPLE: example divrem; shows an example |
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| 134 | " |
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| 135 | { |
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| 136 | return(reduce(f,std(g))); |
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| 137 | } |
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| 138 | example |
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| 139 | { "EXAMPLE:"; echo = 2; |
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| 140 | ring R=0,x,dp; |
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| 141 | divrem(x2+1,x2); |
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| 142 | } |
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| 143 | |
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| 144 | |
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| 145 | //================ chinese remainder theorem for polynomials ================== |
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| 146 | |
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| 147 | proc chinrestp(list b,list moduli) |
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[1a3911] | 148 | "USAGE: chinrestp(b,moduli); moduli, b, moduli=list of polynomials |
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| 149 | RETURN: poly x, s.t. x= b[i] mod moduli[i] |
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[121920] | 150 | NOTE: chinese remainder theorem for polynomials |
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| 151 | EXAMPLE: example chinrestp; shows an example |
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| 152 | " |
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| 153 | { |
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| 154 | int i; |
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| 155 | int n=size(moduli); |
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| 156 | poly M=1; |
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| 157 | for(i=1;i<=n;i++) |
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| 158 | { |
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| 159 | M=M*moduli[i]; |
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| 160 | } |
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| 161 | list m; |
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| 162 | for(i=1;i<=n;i++) |
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| 163 | { |
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| 164 | m[i]=M/moduli[i]; |
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| 165 | } |
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| 166 | list y; |
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| 167 | for(i=1;i<=n;i++) |
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| 168 | { |
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| 169 | y[i]= extgcd(moduli[i],m[i])[3]; |
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| 170 | } |
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| 171 | poly B=0; |
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| 172 | for(i=1;i<=n;i++) |
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| 173 | { |
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| 174 | B=B+y[i]*m[i]*b[i]; |
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| 175 | } |
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| 176 | B=divrem(B,M); |
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| 177 | return(B); |
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| 178 | } |
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| 179 | example |
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| 180 | { "EXAMPLE:"; echo = 2; |
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| 181 | ring R=7,x,dp; |
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| 182 | list b=3x-4, -3x2+1, 1, 4; |
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| 183 | list moduli=(x-2)^2, (x-5)^3, x-1, x-6; |
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| 184 | chinrestp(b,moduli); |
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| 185 | } |
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| 186 | |
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| 187 | |
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| 188 | //========================= norm of a polynomial =============================== |
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| 189 | |
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| 190 | proc norm(poly a, poly b, poly h, poly f) |
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| 191 | "USAGE: norm(a,b,h,f); |
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| 192 | RETURN: norm of a(x)-b(x)y in IF[C] |
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| 193 | NOTE: The norm is a polynomial in just one variable. |
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[1a3911] | 194 | Curve C: y^2+h(x)y=f(x) is defined over basering. |
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[121920] | 195 | EXAMPLE: example norm; shows an example |
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| 196 | " |
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| 197 | { |
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| 198 | poly n=a^2+a*b*h-b^2*f; |
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| 199 | return(n); |
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| 200 | } |
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| 201 | example |
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| 202 | { "EXAMPLE:"; echo = 2; |
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| 203 | ring R=7,x,dp; |
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| 204 | // hyperelliptic curve y^2 + h*y = f |
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| 205 | poly h=x; |
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| 206 | poly f=x5+5x4+6x2+x+3; |
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| 207 | poly a=x2+1; |
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| 208 | poly b=x; |
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| 209 | norm(a,b,h,f); |
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| 210 | } |
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| 211 | |
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| 212 | |
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| 213 | //========== multiplikation of polynomials in the coordinate ring ============= |
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| 214 | |
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| 215 | proc multi(poly a, poly b, poly c, poly d, poly h, poly f) |
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| 216 | "USAGE: multi(a,b,c,d,h,f); |
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| 217 | RETURN: list L with L[1]-L[2]y=(a(x)-b(x)y)*(c(x)-d(x)y) in IF[C] |
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[1a3911] | 218 | NOTE: Curve C: y^2+h(x)y=f(x) is defined over basering. |
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[121920] | 219 | EXAMPLE: example multi; shows an example |
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| 220 | " |
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| 221 | { |
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| 222 | poly A=a*c + b*d*f; |
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| 223 | poly B=b*c +a*d + b*h*d; |
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| 224 | return (list(A,B)); |
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| 225 | } |
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| 226 | example |
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| 227 | { "EXAMPLE:"; echo = 2; |
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| 228 | ring R=7,x,dp; |
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| 229 | poly h=x; |
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| 230 | poly f=x5+5x4+6x2+x+3; |
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| 231 | // hyperelliptic curve y^2 + h*y = f |
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| 232 | poly a=x2+1; |
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| 233 | poly b=x; |
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| 234 | poly c=5; |
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| 235 | poly d=-x; |
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| 236 | multi(a,b,c,d,h,f); |
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| 237 | } |
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| 238 | |
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| 239 | |
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| 240 | //================== polynomial expansion around a point ======================== |
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| 241 | |
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| 242 | proc darst(list P,int k, poly h, poly f) |
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| 243 | "USAGE: darst(P,k,h,f); |
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| 244 | RETURN: list c of length k |
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| 245 | NOTE: expansion around point P in IF[C], s.t. |
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| 246 | y=c[1]+c[2]*(x-P[1]) +...+c[k]*(x-P[1])^k-1 + rest. |
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| 247 | Curve C:y^2+h(x)y=f(x) is defined over basering. |
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| 248 | EXAMPLE: example darst; shows an example |
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| 249 | " |
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| 250 | { |
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| 251 | |
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| 252 | if ( P[2] == -P[2]- subst(h,var(1),P[1])) |
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| 253 | { |
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| 254 | ERROR("no special points allowed"); |
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| 255 | } |
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| 256 | list c; |
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| 257 | list r; |
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| 258 | list n; |
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| 259 | poly N; |
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| 260 | c[1]=P[2]; |
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| 261 | r[1]=list(0,-1,1,0); |
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| 262 | poly r1,r2,r3,r4; |
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| 263 | // rational function are represented as (r1 - r2*y) / (r3 - r4*y) |
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| 264 | |
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| 265 | for (int i=1; i<k ; i++) |
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| 266 | { |
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| 267 | r1=r[i][1]-c[i]*r[i][3]; |
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| 268 | r2=r[i][2]-c[i]*r[i][4]; |
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| 269 | r3=r[i][3]; |
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| 270 | r4=r[i][4]; |
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| 271 | n=multi(r3,r4,r1+r2*h,-r2,h,f); |
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| 272 | N=r1*r1 + r1*r2*h-r2*r2*f; |
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| 273 | r[i+1]=list(N/(var(1)-P[1]),0,n[1],n[2]); |
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| 274 | while ((divrem(r[i+1][1],var(1)-P[1]) ==0) and (divrem(r[i+1][2],var(1)-P[1]) ==0) and (divrem(r[i+1][3],var(1)-P[1]) ==0) and (divrem(r[i+1][4],var(1)-P[1]) ==0)) |
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| 275 | { |
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| 276 | // reducing the rationl function |
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| 277 | //(r[i+1][1] - r[i+1][2]*y)/(r[i+1][3] - r[i+1][4]) , otherwise there |
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| 278 | // could be a pole, s.t. conditions are not fulfilled. |
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| 279 | r[i+1][1]=(r[i+1][1]) / (var(1)-P[1]); |
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| 280 | r[i+1][2]=(r[i+1][2]) / (var(1)-P[1]); |
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| 281 | r[i+1][3]=(r[i+1][3]) / (var(1)-P[1]); |
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| 282 | r[i+1][4]=(r[i+1][4]) / (var(1)-P[1]); |
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| 283 | } |
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| 284 | c[i+1]=(subst(r[i+1][1],var(1),P[1]) - subst(r[i+1][2],var(1),P[1])*P[2]) / (subst(r[i+1][3],var(1),P[1]) - subst(r[i+1][4],var(1),P[1])*P[2]); |
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| 285 | } |
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| 286 | return(c); |
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| 287 | } |
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| 288 | example |
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| 289 | { "EXAMPLE:"; echo = 2; |
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| 290 | ring R=7,x,dp; |
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| 291 | // hyperelliptic curve y^2 + h*y = f |
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| 292 | poly h=x; |
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| 293 | poly f=x5+5x4+6x2+x+3; |
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| 294 | list P=5,3; |
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| 295 | darst(P,3,h,f); |
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| 296 | } |
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| 297 | |
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| 298 | |
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| 299 | //================ rational representation of a divisor ======================= |
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| 300 | |
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| 301 | proc ratrep1 (list P, poly h, poly f) |
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| 302 | "USAGE: ratrep1(P,k,h,f); |
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| 303 | RETURN: list (a,b) |
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| 304 | NOTE: Important: P has to be semireduced! |
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| 305 | Computes rational representation of the divisor |
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| 306 | P[1][3]*(P[1][1], P[1][2]) +...+ P[sizeof(P)][3]* |
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| 307 | *(P[sizeof(P)][1], P[sizeof(P)][2]) - (*)infty=div(a,b) |
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| 308 | Divisor P has to be semireduced. |
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| 309 | Curve C:y^2+h(x)y=f(x) is defined over basering. |
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| 310 | SEE AlSO: ratrep |
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| 311 | EXAMPLE: example ratrep1; shows an example |
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| 312 | " |
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| 313 | { |
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| 314 | poly a=1; |
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| 315 | list b; |
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| 316 | list m; |
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| 317 | list koef; |
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| 318 | int k; |
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| 319 | |
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| 320 | // Determination of the polynomial b[i] for each point using procedure darst |
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| 321 | for (int i=1 ; i<= size(P); i++) |
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| 322 | { |
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| 323 | a=a*(var(1)-P[i][1])^(P[i][3]); // computing polynomial a |
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| 324 | m[i]=(var(1)-P[i][1])^(P[i][3]); |
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| 325 | b[i]=P[i][2]; |
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| 326 | k=1; |
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| 327 | |
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| 328 | while (divrem(b[i]*b[i] + b[i] *h - f,(var(1)-P[i][1])^(P[i][3])) != 0) |
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| 329 | { |
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| 330 | k=k+1; // b[i]=P[i][2]; |
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| 331 | koef=darst(list (P[i][1],P[i][2]), k, h,f); |
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| 332 | // could be improved, if one doesn't compute list coef completely new |
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| 333 | // every time |
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| 334 | b[i]=b[i]+ koef[k]*(var(1)-P[i][1])^(k-1); |
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| 335 | } |
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| 336 | } |
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| 337 | // Return polynomial a and b. Polynomial b is solution of the congruencies |
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| 338 | // b[i] mod m[i] . |
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| 339 | return(list(a,chinrestp(b,m))); |
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| 340 | |
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| 341 | } |
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| 342 | example |
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| 343 | { "EXAMPLE:"; echo = 2; |
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| 344 | ring R=7,x,dp; |
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| 345 | // hyperelliptic curve y^2 + h*y = f |
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| 346 | poly h=x; |
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| 347 | poly f=x5+5x4+6x2+x+3; |
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| 348 | //divisor P |
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| 349 | list P=list(-1,-3,1),list(1,1,1); |
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| 350 | ratrep1(P,h,f); |
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| 351 | } |
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| 352 | |
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| 353 | |
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| 354 | //================ rational representation of a divisor ======================= |
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| 355 | |
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| 356 | proc ratrep (list P, poly h, poly f) |
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| 357 | "USAGE: ratrep(P,k,h,f); |
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| 358 | RETURN: list (a,b) |
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| 359 | NOTE: Importatnt: P has to be semireduced! |
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| 360 | Computes rational representation of the divisor |
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| 361 | P[1][3]*(P[1][1], P[1][2]) +...+ P[sizeof(P)][3]* |
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| 362 | *(P[sizeof(P)][1], P[sizeof(P)][2]) - (*)infty=div(a,b) |
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| 363 | Divisor P has to be semireduced. |
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| 364 | Curve C:y^2+h(x)y=f(x) is defined over basering. |
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| 365 | Works faster than ratrep1. |
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| 366 | SEE ALSO: ratrep1 |
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| 367 | EXAMPLE: example ratrep; shows an example |
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| 368 | " |
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| 369 | { |
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| 370 | poly a=1; |
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| 371 | list b; |
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| 372 | list m; |
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| 373 | list koef; |
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| 374 | int k; |
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| 375 | |
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| 376 | poly c; |
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| 377 | list r; |
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| 378 | list n; |
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| 379 | poly Norm; |
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| 380 | poly r1,r2,r3,r4; |
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| 381 | |
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| 382 | // Determination of the polynomial b[i] for each point using procedure darst |
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| 383 | for (int i=1 ; i<= size(P); i++) |
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| 384 | { |
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| 385 | a=a*(var(1)-P[i][1])^(P[i][3]); // computing polynomial a |
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| 386 | m[i]=(var(1)-P[i][1])^(P[i][3]); |
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| 387 | b[i]=P[i][2]; |
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| 388 | k=1; |
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| 389 | c=P[i][2]; |
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| 390 | r=0,-1,1,0; |
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| 391 | while (divrem(b[i]*b[i] + b[i] *h - f,(var(1)-P[i][1])^(P[i][3])) != 0) |
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| 392 | { |
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| 393 | k=k+1; |
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| 394 | // here, the procedure darst was integrateg. In every pass a new |
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| 395 | // coefficient c[i] is determined. |
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| 396 | r1=r[1]-c*r[3]; |
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| 397 | r2=r[2]-c*r[4]; |
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| 398 | r3=r[3]; |
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| 399 | r4=r[4]; |
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| 400 | n=multi(r3,r4,r1+r2*h,-r2,h,f); |
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| 401 | Norm=r1*r1 + r1*r2*h-r2*r2*f; |
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| 402 | r=list(Norm/(var(1)-P[i][1]),0,n[1],n[2]); |
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| 403 | while ((divrem(r[1],var(1)-P[i][1]) ==0) and (divrem(r[2],var(1)-P[i][1]) ==0) and (divrem(r[3],var(1)-P[i][1]) ==0) and (divrem(r[4],var(1)-P[i][1]) ==0)) |
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| 404 | { |
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| 405 | |
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| 406 | // reducing the rationl function |
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| 407 | // (r[1]-r[2]y)/(r[3]-r[4]y) , otherwise there |
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| 408 | // could be a pole, s.t. conditions are not fulfilled. |
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| 409 | r[1]=(r[1]) / (var(1)-P[i][1]); |
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| 410 | r[2]=(r[2]) / (var(1)-P[i][1]); |
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| 411 | r[3]=(r[3]) / (var(1)-P[i][1]); |
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| 412 | r[4]=(r[4]) / (var(1)-P[i][1]); |
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| 413 | } |
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| 414 | c=(subst(r[1],var(1),P[i][1]) - subst(r[2],var(1),P[i][1])*P[i][2]) / (subst(r[3],var(1),P[i][1]) - subst(r[4],var(1),P[i][1])*P[i][2]); |
---|
| 415 | b[i]=b[i]+ c*(var(1)-P[i][1])^(k-1); |
---|
| 416 | } |
---|
| 417 | } |
---|
| 418 | // return polynomial a and b. Polynomial b is solution of the congruencies |
---|
| 419 | // b[i] mod m[i] . |
---|
| 420 | return(list(a,chinrestp(b,m))); |
---|
| 421 | } |
---|
| 422 | example |
---|
| 423 | { "EXAMPLE:"; echo = 2; |
---|
| 424 | ring R=7,x,dp; |
---|
| 425 | // hyperelliptic curve y^2 + h*y = f |
---|
| 426 | poly h=x; |
---|
| 427 | poly f=x5+5x4+6x2+x+3; |
---|
| 428 | //Divisor P |
---|
| 429 | list P=list(-1,-3,1),list(1,1,1); |
---|
| 430 | ratrep(P,h,f); |
---|
| 431 | } |
---|
| 432 | |
---|
| 433 | |
---|
| 434 | //============== Order of a zero in a polynomial ============================== |
---|
| 435 | |
---|
| 436 | proc ordnung(poly x0 , poly g) |
---|
| 437 | "USAGE: ordnung(x0,g); |
---|
| 438 | RETURN: int i |
---|
| 439 | NOTE: i is maximal, s.t. (x-x0)^i divides g |
---|
| 440 | EXAMPLE: example ordnung; shows an example |
---|
| 441 | " |
---|
| 442 | { |
---|
| 443 | poly gg=g; |
---|
| 444 | int i; |
---|
| 445 | while ( divrem(gg,var(1)-x0) ==0 ) |
---|
| 446 | { |
---|
| 447 | i=i+1; |
---|
| 448 | gg=gg/(var(1)-x0); |
---|
| 449 | } |
---|
| 450 | return(i); |
---|
| 451 | } |
---|
| 452 | example |
---|
| 453 | { "EXAMPLE:"; echo = 2; |
---|
| 454 | ring R=0,x,dp; |
---|
| 455 | poly g=(x-5)^7*(x-3)^2; |
---|
| 456 | number x0=5; |
---|
| 457 | ordnung(x0,g); |
---|
| 458 | } |
---|
| 459 | |
---|
| 460 | |
---|
| 461 | //================== divisor of a polynomial function ========================= |
---|
| 462 | |
---|
| 463 | proc divisor(poly a, poly b, poly h, poly f, list #) |
---|
| 464 | "USAGE: divisor(a,b,h,f); optional: divisor(a,b,h,f,s); s=0,1 |
---|
| 465 | RETURN: list P |
---|
[1a3911] | 466 | NOTE: P[1][3]*(P[1][1], P[1][2]) +...+ P[size(P)][3]* |
---|
| 467 | *(P[size(P)][1], P[size(P)][2]) - (*)infty=div(a(x)-b(x)y) |
---|
[121920] | 468 | if there is an optional parameter s!=0, then divisor additonally |
---|
| 469 | returns a parameter, which says, whether irreducible polynomials |
---|
| 470 | occured during computations or not. Otherwise only warnings are |
---|
| 471 | displayed on the monitor. For s=0 nothing happens. |
---|
[1a3911] | 472 | Curve C: y^2+h(x)y=f(x) is defined over basering. |
---|
[121920] | 473 | EXAMPLE: example divisor; shows an example |
---|
| 474 | " |
---|
| 475 | { |
---|
| 476 | list p; |
---|
| 477 | int j; |
---|
| 478 | poly x0; |
---|
| 479 | list y; |
---|
| 480 | list fa=factorize(gcd(a,b)); // wanted: common roots of a and b |
---|
| 481 | poly Norm=norm(a,b,h,f); |
---|
| 482 | |
---|
| 483 | int s; |
---|
| 484 | int irred=0; |
---|
| 485 | if (size(#)>0) |
---|
| 486 | { |
---|
| 487 | s=#[1]; |
---|
| 488 | } |
---|
| 489 | else |
---|
| 490 | { |
---|
| 491 | s=0; |
---|
| 492 | } |
---|
| 493 | |
---|
| 494 | for (int i=2; i<=size(fa[1]) ; i++) |
---|
| 495 | { |
---|
| 496 | // searching roots by finding polynomials of degree 1 |
---|
| 497 | if ( deg(fa[1][i]) !=1 ) |
---|
| 498 | { |
---|
| 499 | if (s==0) |
---|
| 500 | { |
---|
| 501 | "WARNIG: ", fa[1][i], "is irreducible over this field !"; |
---|
| 502 | } |
---|
| 503 | else |
---|
| 504 | { |
---|
| 505 | irred=1; |
---|
| 506 | } |
---|
| 507 | } |
---|
| 508 | else |
---|
| 509 | { |
---|
| 510 | x0=var(1) - fa[1][i]; |
---|
| 511 | // finding the y-coordinates; max. 2 |
---|
| 512 | y= factorize(var(1)^2 + var(1)*subst(h,var(1),x0) - subst(f,var(1),x0)); |
---|
| 513 | if ( deg(y[1][2]) == 1) |
---|
| 514 | // if root belongs to point on curve, then... |
---|
| 515 | { |
---|
| 516 | // compute order of a-b*y in the founded point |
---|
| 517 | j=j+1; |
---|
| 518 | p[j]=list(x0,var(1)-y[1][2],fa[2][i]); |
---|
| 519 | if ( y[2][2]== 1) // ordinary point |
---|
| 520 | { |
---|
| 521 | j=j+1; |
---|
| 522 | p[j]=list(x0 , var(1)-y[1][3] , fa[2][i] ); |
---|
| 523 | if (a/(var(1)-x0)^(fa[2][i]) - b/(var(1)-x0)^(fa[2][i]) * p[j][2] ==0 ) |
---|
| 524 | { |
---|
| 525 | p[j][3]= p[j][3] + ordnung(x0,norm(a/(var(1)-x0)^(fa[2][i]) , b/(var(1)-x0)^(fa[2][i]),h,f)); |
---|
| 526 | } |
---|
| 527 | if (a/(var(1)-x0)^(fa[2][i]) - b/(var(1)-x0)^(fa[2][i]) * p[j-1][2] ==0 ) |
---|
| 528 | { |
---|
| 529 | p[j-1][3]=p[j-1][3] + ordnung(x0,norm(a/(var(1)-x0)^(fa[2][i]) , b/(var(1)-x0)^(fa[2][i]),h,f)); |
---|
| 530 | } |
---|
| 531 | } |
---|
| 532 | else // special point |
---|
| 533 | { |
---|
| 534 | p[j][3]=p[j][3] *2 ; |
---|
| 535 | if (a/(var(1)-x0)^(fa[2][i]) - b/(var(1)-x0)^(fa[2][i]) * p[j][2] ==0 ) |
---|
| 536 | { |
---|
| 537 | p[j][3]=p[j][3] + ordnung(x0,norm(a/(var(1)-x0)^(fa[2][i]) , b/(var(1)-x0)^(fa[2][i]),h,f)); |
---|
| 538 | } |
---|
| 539 | } |
---|
| 540 | |
---|
| 541 | } |
---|
| 542 | // Norm of a-b*y is reduced by common root of a and b |
---|
| 543 | // (is worked off) |
---|
| 544 | Norm = Norm/((var(1)-x0)^(ordnung(x0,Norm))); |
---|
| 545 | } |
---|
| 546 | } |
---|
| 547 | |
---|
| 548 | // some points are still missing; points for which a and b have no common |
---|
| 549 | // roots, but norm(a-b*Y)=0 . |
---|
| 550 | fa=factorize(Norm); |
---|
| 551 | for ( i=2 ; i<=size(fa[1]) ; i++) |
---|
| 552 | { |
---|
| 553 | if ( deg(fa[1][i]) !=1) |
---|
| 554 | { |
---|
| 555 | if (s==0) |
---|
| 556 | { |
---|
| 557 | "WARNING: ", fa[1][i], "is irreducible over this field !"; |
---|
| 558 | } |
---|
| 559 | else |
---|
| 560 | { |
---|
| 561 | irred=1; |
---|
| 562 | } |
---|
| 563 | } |
---|
| 564 | else |
---|
| 565 | { |
---|
| 566 | x0=var(1) - fa[1][i]; |
---|
| 567 | y= factorize(var(1)^2 + var(1)*subst(h,var(1),x0) - subst(f,var(1),x0)); |
---|
| 568 | if ( deg(y[1][2]) == 1) |
---|
| 569 | // if root belongs to point on curve, then... |
---|
| 570 | { |
---|
| 571 | if (subst(a,var(1),x0)- subst(b,var(1),x0)* (var(1)-y[1][2]) ==0) |
---|
| 572 | { |
---|
| 573 | p[size(p)+1]=list(x0,var(1)-y[1][2], ordnung(x0,Norm,h,f)); |
---|
| 574 | } |
---|
| 575 | if ( y[2][2]== 1) // ordinary point |
---|
| 576 | { |
---|
| 577 | if (subst(a,var(1),x0)- subst(b,var(1),x0)* (var(1)-y[1][3]) ==0) |
---|
| 578 | { |
---|
| 579 | p[size(p)+1]=list(x0 , var(1)-y[1][3] , ordnung(x0,Norm,h,f)); |
---|
| 580 | } |
---|
| 581 | } |
---|
| 582 | } |
---|
| 583 | } |
---|
| 584 | } |
---|
| 585 | if (s==0) |
---|
| 586 | { |
---|
| 587 | return(p); |
---|
| 588 | } |
---|
| 589 | return(p,irred); |
---|
| 590 | } |
---|
| 591 | example |
---|
| 592 | { "EXAMPLE:"; echo = 2; |
---|
| 593 | ring R=7,x,dp; |
---|
| 594 | // hyperelliptic curve y^2 + h*y = f |
---|
| 595 | poly h=x; |
---|
| 596 | poly f=x5+5x4+6x2+x+3; |
---|
| 597 | poly a=(x-1)^2*(x-6); |
---|
| 598 | poly b=0; |
---|
| 599 | divisor(a,b,h,f,1); |
---|
| 600 | } |
---|
| 601 | |
---|
| 602 | |
---|
| 603 | //===================== gcd of two divisors =================================== |
---|
| 604 | |
---|
| 605 | proc gcddivisor(list p, list q) |
---|
| 606 | "USAGE: gcddivisor(p,q); |
---|
| 607 | RETURN: list P |
---|
| 608 | NOTE: gcd of two divisors |
---|
| 609 | EXAMPLE: example gcddivisor; shows an example |
---|
| 610 | " |
---|
| 611 | { |
---|
| 612 | list e; |
---|
| 613 | int i,j; |
---|
| 614 | for (i=1 ; i<= size(p) ; i++) |
---|
| 615 | { |
---|
| 616 | for (j=1 ; j<= size(q) ; j++) |
---|
| 617 | { |
---|
| 618 | if ( p[i][1] == q[j][1] and p[i][2] == q[j][2]) |
---|
| 619 | { |
---|
| 620 | if ( p[i][3] <= q[j][3] ) |
---|
| 621 | { |
---|
| 622 | e[size(e)+1]= list (p[i][1] , p[i][2] , p[i][3]); |
---|
| 623 | } |
---|
| 624 | else |
---|
| 625 | { |
---|
| 626 | e[size(e)+1]= list (q[j][1] , q[j][2] , q[j][3]); |
---|
| 627 | } |
---|
| 628 | } |
---|
| 629 | } |
---|
| 630 | } |
---|
| 631 | return(e); |
---|
| 632 | } |
---|
| 633 | example |
---|
| 634 | { "EXAMPLE:"; echo = 2; |
---|
| 635 | ring R=7,x,dp; |
---|
| 636 | // hyperelliptic curve y^2 + h*y = f |
---|
| 637 | poly h=x; |
---|
| 638 | poly f=x5+5x4+6x2+x+3; |
---|
| 639 | // two divisors |
---|
| 640 | list p=list(-1,-3,1),list(1,1,2); |
---|
| 641 | list q=list(1,1,1),list(2,2,1); |
---|
| 642 | gcddivisor(p,q); |
---|
| 643 | } |
---|
| 644 | |
---|
| 645 | |
---|
| 646 | //========== semireduced divisor from rational representation================= |
---|
| 647 | |
---|
| 648 | proc semidiv(list D,poly h, poly f) |
---|
| 649 | "USAGE: semidiv(D,h,f); |
---|
| 650 | RETURN: list P |
---|
| 651 | NOTE: important: Divisor D has to be semireduced! |
---|
[1a3911] | 652 | Computes semireduced divisor P[1][3]*(P[1][1], P[1][2]) +...+ P[size(P)][3]* |
---|
| 653 | *(P[size(P)][1], P[size(P)][2]) - (*)infty=div(D[1],D[2])@* |
---|
[121920] | 654 | Curve C:y^2+h(x)y=f(x) is defined over basering. |
---|
| 655 | EXAMPLE: example semidiv; shows an example |
---|
| 656 | " |
---|
| 657 | { |
---|
| 658 | if ( deg(D[2]) >= deg(D[1]) or divrem(D[2]^2+D[2]*h-f,D[1]) != 0 ) |
---|
| 659 | { |
---|
| 660 | ERROR("Pair of polynomials doesn't belong to semireduced divisor!"); |
---|
| 661 | } |
---|
| 662 | list D1,D2; |
---|
| 663 | int s1,s2; |
---|
| 664 | D1,s1=divisor(D[1],0,h,f,1); |
---|
| 665 | D2,s2=divisor(D[2],1,h,f,1); |
---|
| 666 | |
---|
| 667 | // Only if irreducible polynomials occured in D1 !and! D2 a warning |
---|
| 668 | // is necessary. |
---|
| 669 | if (s1==1 and s2==1) |
---|
| 670 | { |
---|
| 671 | "Attention: |
---|
| 672 | Perhaps some points were not found over this field!"; |
---|
| 673 | } |
---|
| 674 | return(gcddivisor(D1,D2)); |
---|
| 675 | } |
---|
| 676 | example |
---|
| 677 | { "EXAMPLE:"; echo = 2; |
---|
| 678 | ring R=7,x,dp; |
---|
| 679 | // hyperelliptic curve y^2 + h*y = f |
---|
| 680 | poly h=x; |
---|
| 681 | poly f=x5+5x4+6x2+x+3; |
---|
| 682 | // Divisor |
---|
| 683 | list D=x2-1,2x-1; |
---|
| 684 | semidiv(D,h,f); |
---|
| 685 | } |
---|
| 686 | |
---|
| 687 | |
---|
| 688 | //=============== Cantor's algorithm - composition ============================ |
---|
| 689 | |
---|
| 690 | proc cantoradd(list D, list Q, poly h, poly f) |
---|
| 691 | "USAGE: cantoradd(D,Q,h,f); |
---|
| 692 | RETURN: list P |
---|
| 693 | NOTE: Cantor's Algorithm - composition |
---|
| 694 | important: D and Q have to be semireduced! |
---|
[1a3911] | 695 | Computes semireduced divisor div(P[1],P[2])= div(D[1],D[2]) + div(Q[1],Q[2]) |
---|
[121920] | 696 | The divisors are defined over the basering. |
---|
[1a3911] | 697 | Curve C: y^2+h(x)y=f(x) is defined over the basering. |
---|
[121920] | 698 | EXAMPLE: example cantoradd; shows an example |
---|
| 699 | " |
---|
| 700 | { |
---|
| 701 | poly a; |
---|
| 702 | poly b; |
---|
| 703 | list e=extgcd(D[1],Q[1]); |
---|
| 704 | if ( e[1]==1 ) |
---|
| 705 | { |
---|
| 706 | a=D[1]*Q[1]; |
---|
| 707 | b=divrem( e[2]*D[1]*Q[2] + e[3]*Q[1]*D[2] ,a); |
---|
| 708 | return(list(a,b)); |
---|
| 709 | } |
---|
| 710 | list c=extgcd(e[1],D[2]+Q[2]+h); |
---|
| 711 | poly s1=e[2]*c[2]; |
---|
| 712 | poly s2=c[2]*e[3]; |
---|
| 713 | poly s3=c[3]; |
---|
| 714 | a=D[1]*Q[1]/c[1]^2; |
---|
| 715 | b=divrem((s1*D[1]*Q[2] + s2*Q[1]*D[2] + s3*(D[2]*Q[2] + f))/c[1],a); |
---|
| 716 | return(list(a,b)); |
---|
| 717 | } |
---|
| 718 | example |
---|
| 719 | { "EXAMPLE:"; echo = 2; |
---|
| 720 | ring R=7,x,dp; |
---|
| 721 | // hyperelliptic curve y^2 + h*y = f |
---|
| 722 | poly h=x; |
---|
| 723 | poly f=x5+5x4+6x2+x+3; |
---|
| 724 | // two divisors in rational representation |
---|
| 725 | list D=x2-1,2x-1; |
---|
| 726 | list Q=x2-3x+2,-3x+1; |
---|
| 727 | cantoradd(D,Q,h,f); |
---|
| 728 | } |
---|
| 729 | |
---|
| 730 | |
---|
| 731 | //==================== Cantor's algorithm - reduction ========================= |
---|
| 732 | |
---|
| 733 | proc cantorred(list D,poly h,poly f) |
---|
| 734 | "USAGE: cantorred(D,h,f); |
---|
| 735 | RETURN: list N |
---|
| 736 | NOTE: Cantor's algorithm - reduction. |
---|
| 737 | important: Divisor D has to be semireduced! |
---|
[1a3911] | 738 | Computes reduced divisor div(N[1],N[2])= div(D[1],D[2]).@* |
---|
[121920] | 739 | The divisors are defined over the basering. |
---|
[1a3911] | 740 | Curve C: y^2+h(x)y=f(x) is defined over the basering. |
---|
[121920] | 741 | EXAMPLE: example cantorred; shows an example |
---|
| 742 | " |
---|
| 743 | { |
---|
| 744 | list N=D; |
---|
| 745 | while ( 2*deg(N[1]) > deg(f)-1 ) |
---|
| 746 | { |
---|
| 747 | N[1]=(f - N[2]*h - N[2]^2)/N[1]; |
---|
| 748 | N[2]=divrem(-h-N[2],N[1]); |
---|
| 749 | } |
---|
| 750 | N[1]=N[1]/leadcoef(N[1]); |
---|
| 751 | return(N); |
---|
| 752 | } |
---|
| 753 | example |
---|
| 754 | { "EXAMPLE:"; echo = 2; |
---|
| 755 | ring R=7,x,dp; |
---|
| 756 | // hyperelliptic curve y^2 + h*y = f |
---|
| 757 | poly h=x; |
---|
| 758 | poly f=x5+5x4+6x2+x+3; |
---|
| 759 | // semireduced divisor |
---|
| 760 | list D=2x4+3x3-3x-2, -x3-2x2+3x+1; |
---|
| 761 | cantorred(D,h,f); |
---|
| 762 | } |
---|
| 763 | |
---|
| 764 | |
---|
| 765 | //================= doubling a semireduced divisor ============================ |
---|
| 766 | |
---|
| 767 | proc double(list D, poly h, poly f) |
---|
| 768 | "USAGE: double(D,h,f); |
---|
| 769 | RETURN: list Q=2*D |
---|
| 770 | NOTE: important: Divisor D has to be semireduced! |
---|
| 771 | Special case of Cantor's algorithm. |
---|
[1a3911] | 772 | Computes reduced divisor div(Q[1],Q[2])= 2*div(D[1],D[2]).@* |
---|
[121920] | 773 | The divisors are defined over the basering. |
---|
| 774 | Curve C:y^2+h(x)y=f(x) is defined over the basering. |
---|
| 775 | EXAMPLE: example double; shows an example |
---|
| 776 | " |
---|
| 777 | { |
---|
| 778 | list c=extgcd(D[1], 2*D[2] + h); |
---|
| 779 | poly a=D[1]*D[1]/c[1]^2; |
---|
| 780 | poly b=divrem((c[2]*D[1]*D[2] + c[3]*(D[2]*D[2] + f))/c[1],a); |
---|
| 781 | return(cantorred(list(a,b),h,f)); |
---|
| 782 | } |
---|
| 783 | example |
---|
| 784 | { "EXAMPLE:"; echo = 2; |
---|
| 785 | ring R=7,x,dp; |
---|
| 786 | // hyperelliptic curve y^2 + h*y = f |
---|
| 787 | poly h=x; |
---|
| 788 | poly f=x5+5x4+6x2+x+3; |
---|
| 789 | // reduced divisor |
---|
| 790 | list D=x2-1,2x-1; |
---|
| 791 | double(D,h,f); |
---|
| 792 | } |
---|
| 793 | |
---|
| 794 | |
---|
| 795 | //================ multiples of a semireduced divisor ========================= |
---|
| 796 | |
---|
| 797 | proc cantormult(int m, list D, poly h, poly f) |
---|
| 798 | "USAGE: cantormult(m,D,h,f); |
---|
| 799 | RETURN: list res=m*D |
---|
| 800 | NOTE: important: Divisor D has to be semireduced! |
---|
| 801 | Uses repeated doublings for a faster computation |
---|
| 802 | of the reduced divisor m*D. |
---|
[1a3911] | 803 | Attention: Factor m=int, this means bounded. |
---|
[121920] | 804 | For m<0 the inverse of m*D is returned. |
---|
| 805 | The divisors are defined over the basering. |
---|
[1a3911] | 806 | Curve C: y^2+h(x)y=f(x) is defined over the basering. |
---|
[121920] | 807 | EXAMPLE: example cantormult; shows an example |
---|
| 808 | " |
---|
| 809 | { |
---|
| 810 | list res=1,0; |
---|
| 811 | list bas=D; |
---|
| 812 | int exp=m; |
---|
| 813 | if (exp==0) { return(list(1,0)); } |
---|
| 814 | if (exp==1) { return(D); } |
---|
| 815 | if (exp==-1) { return(list(D[1],-D[2]-h)) ; } |
---|
| 816 | if ( exp < 0) |
---|
| 817 | { |
---|
| 818 | exp=-exp; |
---|
| 819 | } |
---|
| 820 | while ( exp > 0 ) |
---|
| 821 | { |
---|
| 822 | if ( (exp mod 2) !=0 ) |
---|
| 823 | { |
---|
| 824 | res = cantorred(cantoradd(res,bas,h,f),h,f); |
---|
| 825 | exp=exp-1; |
---|
| 826 | } |
---|
| 827 | bas=double(bas,h,f); |
---|
[1f9a84] | 828 | exp=exp div 2; |
---|
[121920] | 829 | } |
---|
| 830 | if ( m < 0 ) |
---|
| 831 | { |
---|
| 832 | res[2]=-res[2]-h; |
---|
| 833 | } |
---|
| 834 | return(res); |
---|
| 835 | } |
---|
| 836 | example |
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| 837 | { "EXAMPLE:"; echo = 2; |
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| 838 | ring R=7,x,dp; |
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| 839 | // hyperelliptic curve y^2 + h*y = f |
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| 840 | poly h=x; |
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| 841 | poly f=x5+5x4+6x2+x+3; |
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| 842 | // reduced divisor |
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| 843 | list D=x2-1,2x-1; |
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| 844 | cantormult(34,D,h,f); |
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| 845 | } |
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| 846 | |
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| 847 | |
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| 848 | /* |
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| 849 | //============================================================================= |
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| 850 | // In the following you find a large example, which demostrates the use of |
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| 851 | // the most important procedures. |
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| 852 | //============================================================================= |
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| 853 | //---- field with 2^5=32 elements ---- |
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| 854 | ring r=(2,a),x,dp; |
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| 855 | minpoly=a5+a2+1; |
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| 856 | |
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| 857 | //---- hyperelliptic curve y^2 + hy = f ---- |
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| 858 | poly h=x2+x; |
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| 859 | poly f=x5+x3+1; |
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| 860 | |
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| 861 | //---- two divisors ---- |
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| 862 | list l1=list(a30,0,1),list(0,1,1); |
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| 863 | list l2=list(a30,a16,1),list(1,1,1); |
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| 864 | |
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| 865 | //---- their rational representation ---- |
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| 866 | list D1=ratrep(l1,h,f); D1; |
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| 867 | //[1]: |
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| 868 | // x2+(a4+a)*x |
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| 869 | //[2]: |
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| 870 | // (a)*x+1 |
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| 871 | |
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| 872 | list D2=ratrep(l2,h,f); D2; |
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| 873 | //[1]: |
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| 874 | // x2+(a4+a+1)*x+(a4+a) |
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| 875 | //[2]: |
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| 876 | // (a3+a2+a+1)*x+(a3+a2+a) |
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| 877 | |
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| 878 | //---- back to the point-based-representation ---- |
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| 879 | semidiv(D1,h,f); |
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| 880 | //[1]: |
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| 881 | // [1]: |
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| 882 | // (a4+a) |
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| 883 | // [2]: |
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| 884 | // 0 |
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| 885 | // [3]: |
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| 886 | // 1 |
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| 887 | //[2]: |
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| 888 | // [1]: |
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| 889 | // 0 |
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| 890 | // [2]: |
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| 891 | // 1 |
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| 892 | // [3]: |
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| 893 | // 1 |
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| 894 | |
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| 895 | semidiv(D2,h,f); |
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| 896 | //[1]: |
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| 897 | // [1]: |
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| 898 | // (a4+a) |
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| 899 | // [2]: |
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| 900 | // (a4+a3+a+1) |
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| 901 | // [3]: |
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| 902 | // 1 |
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| 903 | //[2]: |
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| 904 | // [1]: |
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| 905 | // 1 |
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| 906 | // [2]: |
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| 907 | // 1 |
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| 908 | // [3]: |
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| 909 | // 1 |
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| 910 | |
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| 911 | //---- adding D1 and D2 ---- |
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| 912 | list D12=cantorred(cantoradd(D1,D2,h,f),h,f); D12; |
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| 913 | //[1]: |
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| 914 | // x2+x |
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| 915 | //[2]: |
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| 916 | // 1 |
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| 917 | |
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| 918 | //---- D1+D2 in point-based-representation ---- |
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| 919 | semidiv(D12,h,f); |
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| 920 | //[1]: |
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| 921 | // [1]: |
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| 922 | // 1 |
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| 923 | // [2]: |
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| 924 | // 1 |
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| 925 | // [3]: |
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| 926 | // 1 |
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| 927 | //[2]: |
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| 928 | // [1]: |
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| 929 | // 0 |
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| 930 | // [2]: |
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| 931 | // 1 |
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| 932 | // [3]: |
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| 933 | // 1 |
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| 934 | |
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| 935 | //---- D1 + D1 (2 possible ways, same result) ---- |
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| 936 | cantorred(cantoradd(D1,D1,h,f),h,f); |
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| 937 | double(D1,h,f); |
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| 938 | //[1]: |
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| 939 | // x2+(a3+1) |
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| 940 | //[2]: |
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| 941 | // (a4+a3+a+1)*x+(a4+a3+a2+a+1) |
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| 942 | |
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| 943 | //---- order of D1 in the jacobian over the basering ---- |
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| 944 | int i=1; |
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| 945 | list E=D1; |
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| 946 | while (E[1] != 1 or E[2] != 0 ) |
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| 947 | { |
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| 948 | E= cantorred(cantoradd(E,D1,h,f),h,f); |
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| 949 | i=i+1; |
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| 950 | } |
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| 951 | i; // 482 |
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| 952 | |
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| 953 | //---- proof with multiplikation validates the result ---- |
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| 954 | cantormult(i,D1,h,f); |
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| 955 | //[1]: |
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| 956 | // 1 |
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| 957 | //[2]: |
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| 958 | // 0 |
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| 959 | |
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| 960 | //---- computing the inverse of D1 ---- |
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| 961 | list d1= cantormult(-1,D1,h,f); d1; |
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| 962 | //[1]: |
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| 963 | // x2+(a4+a)*x |
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| 964 | //[2]: |
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| 965 | // x2+(a+1)*x+1 |
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| 966 | |
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| 967 | //---- proof validates the result ---- |
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| 968 | cantoradd(d1,D1,h,f); |
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| 969 | //[1]: |
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| 970 | // 1 |
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| 971 | //[2]: |
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| 972 | // 0 |
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| 973 | |
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| 974 | |
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| 975 | */ |
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