[380a17b] | 1 | //////////////////////////////////////////////////////////////////////////// |
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[3686937] | 2 | version="version reesclos.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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[2347b7] | 3 | category="Commutative Algebra"; |
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[845282] | 4 | |
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[9e1207] | 5 | info=" |
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[845282] | 6 | LIBRARY: reesclos.lib PROCEDURES TO COMPUTE THE INT. CLOSURE OF AN IDEAL |
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[f0df61e] | 7 | AUTHOR: Tobias Hirsch, email: hirsch@math.tu-cottbus.de |
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[89b9ee] | 8 | Janko Boehm, email: boehm@mathematik.uni-kl.de |
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| 9 | Magdaleen Marais, email: magdaleen@aims.ac.za |
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[f0df61e] | 10 | |
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| 11 | OVERVIEW: |
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| 12 | A library to compute the integral closure of an ideal I in a polynomial ring |
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| 13 | R=k[x(1),...,x(n)] using the Rees Algebra R[It] of I. It computes the integral |
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[89b9ee] | 14 | closure of R[It], |
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[f0df61e] | 15 | which is a graded subalgebra of R[t]. The degree-k-component is the integral |
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| 16 | closure of the k-th power of I. |
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| 17 | |
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[89b9ee] | 18 | In contrast to the previous version, the library uses 'normal.lib' to compute the |
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| 19 | integral closure of R[It]. This improves the performance considerably. |
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| 20 | |
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[b9b906] | 21 | PROCEDURES: |
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[50cbdc] | 22 | ReesAlgebra(I); computes the Rees Algebra of an ideal I |
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| 23 | normalI(I[,p[,r]]); computes the integral closure of an ideal I using R[It] |
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[9e1207] | 24 | "; |
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| 25 | |
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[50cbdc] | 26 | LIB "normal.lib"; // for HomJJ |
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[f0df61e] | 27 | LIB "standard.lib"; // for groebner |
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[9e1207] | 28 | |
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| 29 | /////////////////////////////////////////////////////////////////////////////// |
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| 30 | |
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| 31 | proc ReesAlgebra (ideal I) |
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[f0df61e] | 32 | "USAGE: ReesAlgebra (I); I = ideal |
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[845282] | 33 | RETURN: The Rees algebra R[It] as an affine ring, where I is an ideal in R. |
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| 34 | The procedure returns a list containing two rings: |
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| 35 | [1]: a ring, say RR; in the ring an ideal ker such that R[It]=RR/ker |
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| 36 | |
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| 37 | [2]: a ring, say Kxt; the basering with additional variable t |
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| 38 | containing an ideal mapI that defines the map RR-->Kxt |
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[9e1207] | 39 | EXAMPLE: example ReesAlgebra; shows an example |
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| 40 | " |
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| 41 | { |
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| 42 | // remember the data of the basering |
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| 43 | |
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| 44 | def oldring = basering; |
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| 45 | string oldchar = charstr(basering); |
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| 46 | string oldvar = varstr(basering); |
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| 47 | string oldord = ordstr(basering); |
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| 48 | int n = ncols(I); |
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| 49 | ideal m = maxideal(1); |
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| 50 | |
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[50cbdc] | 51 | |
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[9e1207] | 52 | // Create a new ring with variables for each generator of I |
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| 53 | |
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| 54 | execute ("ring Rees = "+oldchar+",("+oldvar+",U(1.."+string(n)+")),dp"); |
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| 55 | |
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[50cbdc] | 56 | |
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[9e1207] | 57 | // Kxt is the old ring with additional variable t |
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| 58 | // Here I -> t*I, so the generators of I generate the subalgebra R[It] in Kxt |
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| 59 | |
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| 60 | execute ("ring Kxt = "+oldchar+",("+oldvar+",t),dp"); |
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| 61 | ideal I = fetch(oldring,I); |
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| 62 | ideal m = fetch(oldring,m); |
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| 63 | int k; |
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| 64 | for (k=1;k<=n;k++) |
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| 65 | { |
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[50cbdc] | 66 | I[k]=t*I[k]; |
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[9e1207] | 67 | } |
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[50cbdc] | 68 | |
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[9e1207] | 69 | |
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| 70 | // Now we map from Rees to Kxt, identity on the original variables, and |
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| 71 | // U(k) -> I[k] |
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| 72 | |
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| 73 | ideal mapI = m,I; |
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| 74 | map phi = Rees,mapI; |
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| 75 | ideal zero = 0; |
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| 76 | export (mapI); |
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| 77 | |
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| 78 | // Now the Rees-Algebra is Rees/ker(phi) |
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| 79 | |
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| 80 | setring Rees; |
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| 81 | ideal ker = preimage(Kxt,phi,zero); |
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| 82 | export (ker); |
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[50cbdc] | 83 | |
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[9e1207] | 84 | list result = Rees,Kxt; |
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[50cbdc] | 85 | |
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[9e1207] | 86 | return(result); |
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[50cbdc] | 87 | |
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[9e1207] | 88 | } |
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| 89 | example |
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| 90 | { |
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[f0df61e] | 91 | "EXAMPLE:"; echo=2; |
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[9e1207] | 92 | ring R = 0,(x,y),dp; |
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| 93 | ideal I = x2,xy4,y5; |
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| 94 | list L = ReesAlgebra(I); |
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[845282] | 95 | def Rees = L[1]; // defines the ring Rees, containing the ideal ker |
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| 96 | setring Rees; // passes to the ring Rees |
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[9e1207] | 97 | Rees; |
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[845282] | 98 | ker; // R[It] is isomorphic to Rees/ker |
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[9e1207] | 99 | } |
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[50cbdc] | 100 | |
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[9e1207] | 101 | |
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| 102 | //////////////////////////////////////////////////////////////////////////// |
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| 103 | |
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[845282] | 104 | static |
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| 105 | proc ClosureRees (list L) |
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[50cbdc] | 106 | "USAGE: ClosureRees (L); L a list |
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[845282] | 107 | ASSUME: L is a list containing |
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[50cbdc] | 108 | - a ring L[1], inside L[1] an ideal ker such that L[1]/ker is |
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[9e1207] | 109 | isomorphic to the Rees Algebra R[It] of an ideal I in k[x] |
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| 110 | - a ring L[2]=k[x,t], inside L[1] an ideal mapI defining the |
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| 111 | map L[1] --> L[2] with image R[It] |
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[845282] | 112 | RETURN: quotients of elements of k[x,t] representing generators of the |
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[50cbdc] | 113 | integral closure of R[It]. The result of ClosureRees is a list |
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[845282] | 114 | images, the first size(images)-1 entries are the numerators of the |
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| 115 | generators, the last one is the universal denominator |
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[9e1207] | 116 | " |
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| 117 | { |
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[50cbdc] | 118 | int dblvl=printlevel-voice+2; // toggles how much data is printed |
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[9e1207] | 119 | // during the procedure |
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| 120 | |
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| 121 | def Kxt = basering; |
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| 122 | def R(1) = L[1]; |
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| 123 | setring R(1); // declaration of variables used later |
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| 124 | ideal ker(1)=ker; // in STEP 2 |
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[89b9ee] | 125 | list nor = normal(ker); |
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| 126 | list preimages=nor[2]; |
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[9e1207] | 127 | setring Kxt; |
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| 128 | map psi=R(1),mapI; // from ReesAlgebra: the map Rees->Kxt |
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[89b9ee] | 129 | ideal images=(psi(preimages))[1]; |
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[f84dfa] | 130 | ideal psii = images[size(images)]*ideal(psi); |
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[89b9ee] | 131 | list imagesl = images[1..size(images)]; |
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[f84dfa] | 132 | list psil =psii[1..size(psii)]; |
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| 133 | imagesl=psil+imagesl; |
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[89b9ee] | 134 | return(imagesl); |
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[9e1207] | 135 | } |
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| 136 | |
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| 137 | |
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| 138 | //////////////////////////////////////////////////////////////////////////// |
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| 139 | |
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[845282] | 140 | static |
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| 141 | proc ClosurePower(list images, list #) |
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| 142 | "USAGE: ClosurePower (L [,#]); L a list, # an optional list containing an |
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| 143 | integer |
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| 144 | ASSUME: - L is a list containing generators of the closure of R[It] in k[x,t] |
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[9e1207] | 145 | (the first size(L)-1 elements are the numerators, the last one |
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| 146 | is the denominator) |
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[50cbdc] | 147 | - if # is given: #[1] is an integer, compute generators for the |
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[9e1207] | 148 | closure of I, I^2, ..., I^#[1] |
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[845282] | 149 | RETURN: the integral closure of I, ... I^#[1]. If # is not given, compute |
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[9e1207] | 150 | the closure of all powers up to the maximum degree in t occurring |
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[845282] | 151 | in the closure of R[It] (so this is the last power whose closure is |
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| 152 | not just the sum/product of the smaller powers). The returned |
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| 153 | result is a list of elements of k[x,t] containing generators of the |
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| 154 | closure of the desired powers of I. " |
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[9e1207] | 155 | { |
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[50cbdc] | 156 | int dblvl=printlevel-voice+2; // toggles how much data is printed |
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| 157 | // during the procedure |
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[9e1207] | 158 | |
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| 159 | int j,k,d,computepow; // some counters |
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| 160 | int pow=0; |
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| 161 | int length = size(images)-1; // the number of generators |
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| 162 | poly image; |
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[6f84e21] | 163 | poly @denominator = images[length+1]; // the universal denominator |
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[9e1207] | 164 | |
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| 165 | if (size(#)>0) |
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| 166 | { |
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| 167 | pow=#[1]; |
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| 168 | } |
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| 169 | computepow=pow; |
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| 170 | |
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| 171 | if (dblvl>0) |
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| 172 | { |
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| 173 | ""; |
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| 174 | "// The generators of the closure of R[It]:"; |
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| 175 | } |
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| 176 | |
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| 177 | intmat m[nvars(basering)-1][1]; // an intvec used for jet and maxdeg1 |
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[50cbdc] | 178 | intvec tw=m,1; // such that t has weight 1 and all |
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[9e1207] | 179 | // other variables have weight 0 |
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| 180 | |
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| 181 | // Construct the generators of the closure of R[It] as elements of k[x,t] |
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| 182 | // If # is not given, determine the highest degree pow in t that occurs. |
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| 183 | |
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| 184 | for (j=1;j<=length;j++) |
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| 185 | { |
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[6f84e21] | 186 | images[j] = (images[j]/@denominator); // construct the fraction |
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[9e1207] | 187 | image = images[j]; |
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| 188 | if (dblvl>0) |
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| 189 | { |
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| 190 | "generator",j,":",image; |
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| 191 | } |
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| 192 | |
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| 193 | if (computepow==0) // #[1] not given or ==0 => compute pow |
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| 194 | { |
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| 195 | if (maxdeg1(image,tw)>pow) // from poly.lib |
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| 196 | { |
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| 197 | pow=maxdeg1(image,tw); |
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| 198 | } |
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| 199 | } |
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[50cbdc] | 200 | } |
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[9e1207] | 201 | |
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| 202 | if (dblvl>0) |
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| 203 | { |
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| 204 | ""; |
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| 205 | if (computepow==0) |
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| 206 | { |
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| 207 | "// Compute the closure up to the given powers of I"; |
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| 208 | } |
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| 209 | else |
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| 210 | { |
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| 211 | "// Compute the closure up to the maximal power of t that occured:",pow; |
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| 212 | } |
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| 213 | } |
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| 214 | |
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| 215 | // Construct a list consisting of #[1] resp. pow times the zero ideal |
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| 216 | |
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| 217 | ideal CurrentPower=0; |
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| 218 | list result; |
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| 219 | for (k=1;k<=pow;k++) |
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| 220 | { |
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| 221 | result=insert(result,CurrentPower); |
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| 222 | } |
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| 223 | |
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| 224 | // For each generator and each k, add its degree-k-coefficient to the # |
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| 225 | // closure of I^k |
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| 226 | |
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| 227 | for (j=1;j<=length;j++) |
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| 228 | { |
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| 229 | for (k=1;k<=pow;k++) |
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| 230 | { |
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| 231 | image=images[j]-jet(images[j],k-1,tw); |
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[50cbdc] | 232 | if (image<>0) |
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| 233 | { |
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[9e1207] | 234 | image=subst(image/t^k,t,0); |
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| 235 | if (image<>0) |
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| 236 | { |
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| 237 | result[k]=result[k]+image; |
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| 238 | } |
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| 239 | } |
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| 240 | } |
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| 241 | } |
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| 242 | |
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| 243 | if (dblvl>0) |
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| 244 | { |
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| 245 | ""; |
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| 246 | "// The 'pure' parts of degrees 1..pow:"; |
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| 247 | result; |
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| 248 | ""; |
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| 249 | } |
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| 250 | |
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| 251 | // finally, add the suitable products of generators in lower degrees |
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| 252 | |
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| 253 | for (k=2;k<=pow;k++) |
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[50cbdc] | 254 | { |
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[9e1207] | 255 | for (j=1;j<=(k div 2);j++) |
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[50cbdc] | 256 | { |
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[9e1207] | 257 | result[k]=result[k]+result[j]*result[k-j]; |
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| 258 | } |
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| 259 | } |
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[50cbdc] | 260 | |
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[9e1207] | 261 | return(result); |
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| 262 | } |
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| 263 | |
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| 264 | //////////////////////////////////////////////////////////////////////////// |
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| 265 | |
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| 266 | proc normalI(ideal I, list #) |
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[f0df61e] | 267 | "USAGE: normalI (I [,p[,r]]); I an ideal, p and r optional integers |
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[845282] | 268 | RETURN: the integral closure of I, ..., I^p, where I is an ideal in the |
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| 269 | polynomial ring R=k[x(1),...x(n)]. If p is not given, or p==0, |
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[b9b906] | 270 | compute the closure of all powers up to the maximum degree in t |
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[845282] | 271 | occurring in the closure of R[It] (so this is the last power whose |
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| 272 | closure is not just the sum/product of the smaller). If r |
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| 273 | is given and r==1, normalI starts with a check whether I is already a |
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| 274 | radical ideal. |
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| 275 | The result is a list containing the closure of the desired powers of |
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| 276 | I as ideals of the basering. |
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| 277 | DISPLAY: The procedure displays more comments for higher printlevel. |
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[f0df61e] | 278 | EXAMPLE: example normalI; shows an example |
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[9e1207] | 279 | " |
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| 280 | { |
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[b9b906] | 281 | int dblvl=printlevel-voice+2; // toggles how much data is printed |
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[9e1207] | 282 | // during the procedure |
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| 283 | |
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| 284 | def BAS=basering; // remember the basering |
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| 285 | |
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| 286 | // two simple cases: principal ideals and radical ideals are always |
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| 287 | // integrally closed |
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| 288 | |
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| 289 | if (size(I)<=1) // includes the case I=(0) |
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| 290 | { |
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| 291 | if (dblvl>0) |
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| 292 | { |
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| 293 | "// Trivial case: I is a principal ideal"; |
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[50cbdc] | 294 | } |
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[9e1207] | 295 | list result=I; |
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| 296 | if (size(#)>0) |
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| 297 | { |
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| 298 | for (int k=1;k<=#[1]-1;k++) |
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| 299 | { |
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| 300 | result=insert(result,I*result[k],k); |
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| 301 | } |
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[50cbdc] | 302 | } |
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[9e1207] | 303 | return(result); |
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| 304 | } |
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| 305 | |
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[845282] | 306 | int testrad=0; // do the radical check? |
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[9e1207] | 307 | if (size(#)>1) |
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| 308 | { |
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[845282] | 309 | testrad=#[2]; |
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[9e1207] | 310 | } |
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[845282] | 311 | |
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[9e1207] | 312 | if (testrad==1) |
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| 313 | { |
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| 314 | if (dblvl>0) |
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| 315 | { |
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| 316 | "//Check whether I is radical"; |
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| 317 | } |
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| 318 | |
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| 319 | if (size(reduce(radical(I),std(I)))==0) |
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| 320 | { |
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| 321 | if (dblvl>0) |
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| 322 | { |
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| 323 | "//Trivial case: I is a radical ideal"; |
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| 324 | } |
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| 325 | list result=I; |
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| 326 | if (size(#)>0) |
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| 327 | { |
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| 328 | for (int k=1;k<=#[1]-1;k++) |
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| 329 | { |
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| 330 | result=insert(result,I*result[k],k); |
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| 331 | } |
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[50cbdc] | 332 | } |
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[9e1207] | 333 | return(result); |
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| 334 | } |
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| 335 | } |
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| 336 | |
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| 337 | // start with the computation of the Rees Algebra R[It] of I |
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| 338 | |
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| 339 | if (dblvl>0) |
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| 340 | { |
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| 341 | "// We start with the Rees Algebra of I:"; |
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| 342 | } |
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[50cbdc] | 343 | |
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[9e1207] | 344 | list Rees = ReesAlgebra(I); |
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| 345 | def R(1)=Rees[1]; |
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| 346 | def Kxt=Rees[2]; |
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| 347 | setring R(1); |
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[50cbdc] | 348 | |
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[9e1207] | 349 | if (dblvl>0) |
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| 350 | { |
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| 351 | R(1); |
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| 352 | ker; |
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| 353 | ""; |
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| 354 | "// Now ClosureRees computes generators for the integral closure"; |
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| 355 | "// of R[It] step by step"; |
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[50cbdc] | 356 | } |
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[9e1207] | 357 | |
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| 358 | // ClosureRees computes fractions in R[x,t] representing the generators |
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| 359 | // of the closure of R[It] in k[x,t], which is the same as the closure |
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| 360 | // in Q(R[It]). |
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| 361 | // the first size(images)-1 entries are the numerators of the gene- |
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| 362 | // rators, the last entry is the 'universal' denominator |
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| 363 | |
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| 364 | setring Kxt; |
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| 365 | list images = ClosureRees(Rees); |
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| 366 | |
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| 367 | // ClosureRees was done after the first HomJJ-call |
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| 368 | // ==> I is integrally closed, and images consists of the only entry "closed" |
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| 369 | |
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| 370 | if ((size(images)==1) && (typeof(images[1])=="string")) |
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| 371 | { |
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| 372 | if (dblvl>0) |
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| 373 | { |
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| 374 | "//I is integrally closed!"; |
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| 375 | } |
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[50cbdc] | 376 | |
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[9e1207] | 377 | setring BAS; |
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| 378 | list result=I; |
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| 379 | if (size(#)>0) |
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| 380 | { |
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| 381 | for (int k=1;k<=#[1]-1;k++) |
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| 382 | { |
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| 383 | result=insert(result,I*result[k],k); |
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| 384 | } |
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[50cbdc] | 385 | } |
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[9e1207] | 386 | return(result); |
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| 387 | } |
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[50cbdc] | 388 | |
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[9e1207] | 389 | // construct the fractions corresponding to the generators of the |
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[50cbdc] | 390 | // closure of I and its powers, depending on # (in fact, they will |
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[9e1207] | 391 | // not be real fractions, of course). This is done in ClosurePower. |
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| 392 | list result = ClosurePower(images,#); |
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| 393 | |
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| 394 | // finally fetch the result to the old basering |
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| 395 | |
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| 396 | setring BAS; |
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| 397 | list result=fetch(Kxt,result); |
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[50cbdc] | 398 | return(result); |
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[9e1207] | 399 | } |
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| 400 | example |
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| 401 | { |
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[f0df61e] | 402 | "EXAMPLE:"; echo=2; |
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[9e1207] | 403 | ring R=0,(x,y),dp; |
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| 404 | ideal I = x2,xy4,y5; |
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| 405 | list J = normalI(I); |
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| 406 | I; |
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[845282] | 407 | J; // J[1] is the integral closure of I |
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[9e1207] | 408 | } |
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[845282] | 409 | |
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[8942a5] | 410 | /* |
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[845282] | 411 | LIB"reesclos.lib"; |
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| 412 | |
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| 413 | |
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| 414 | // 1. x^i,y^i in k[x,y] |
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| 415 | // geht bis i = 19 (800sec), bis i=10 wenige Sekunden, |
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| 416 | // bei i = 20 ueber 1GB Hauptspeicher, in der 9. Iteration no memory |
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| 417 | // (braucht 20 Iterationen) |
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| 418 | |
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| 419 | ring r = 0,(x,y),dp; |
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| 420 | int i = 6; |
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| 421 | ideal I = x^i,y^i; |
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| 422 | list J = normalI(I); |
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| 423 | I; |
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| 424 | J; |
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| 425 | |
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| 426 | |
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| 427 | //================================================================ |
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| 428 | |
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| 429 | |
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| 430 | // 2. x^i,y^i,z^i in k[x,y,z] |
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[0610f0e] | 431 | // aehnlich wie 1., funktioniert aber nur bis i=5 und dauert dort |
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[845282] | 432 | // >1 h |
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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 | |
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| 438 | // 3. scheitert in der ersten Iteration beim Radikal |
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| 439 | // Standardbasis des singulaeren Ortes: 7h (in char0), |
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| 440 | // in char(p) viel schneller, obwohl kleine Koeffizienten |
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| 441 | // schon bei Radikal -Test braucht er zu lang (>1h) |
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| 442 | |
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| 443 | ring r = 0,(x,y,z),dp; |
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| 444 | //ring r = 32003,(x,y,z),dp; |
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| 445 | |
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| 446 | ideal I = x2+xy3-5z,z3+y2-xzy,x2y3z5+y3-y5; |
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| 447 | list l= ReesAlgebra(I); |
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| 448 | list J = normalI(I); |
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| 449 | I; |
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| 450 | J; |
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| 451 | |
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[8942a5] | 452 | */ |
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[845282] | 453 | |
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