1 | LIB "tst.lib"; |
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2 | tst_init(); |
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3 | |
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4 | proc finitenessTest(ideal I) |
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5 | "USAGE: finitenessTest(ideal I) |
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6 | RETURN: A list l, l[1] is 1 or 0 and l[2] is an ideal gener- |
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7 | ated by a subset of the variables. l[1]=1 if the map |
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8 | basering/I <-- K[l[2]] is finite and 0 else. |
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9 | NOTE: It is assumed that I is a reduced standard basis |
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10 | with respect to the lexicographical ordering lp, |
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11 | sorted w.r.t. increasing leading terms. |
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12 | " |
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13 | { |
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14 | intvec w=leadexp(I[1]); |
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15 | int j,t; |
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16 | int s=1; |
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17 | ideal k; |
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18 | //----------- check leading exponents ---------------------- |
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19 | //compute s such that lead(I[1]) depends only on |
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20 | //var(s),...,var(n) by inspection of the leading exponents |
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21 | while (w[s]==0) {s++;} |
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22 | for (j=1; j<= size(I); j++) |
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23 | { |
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24 | w=leadexp(I[j]); |
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25 | if (size(ideal(w))==1) {t++;} |
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26 | } |
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27 | //----------------check finiteness ------------------------- |
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28 | //t is the number of elements of the standard basis which |
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29 | //have pure powers in the variables var(1),...,var(s) as |
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30 | //leading term. The map is finite iff s=t. |
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31 | if(s!=t) {return(list(0,k));} |
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32 | for (j=s+1; j<= nvars(basering);j++) |
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33 | { |
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34 | k[j]=var(j); |
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35 | } |
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36 | return (list(1,k)); |
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37 | } |
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38 | |
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39 | |
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40 | proc noetherNormal(ideal id) |
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41 | "USAGE: noetherNormal(id); id ideal |
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42 | RETURN: two ideals i1,i2, where i2 is given by a subset of |
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43 | the variables and i1 defines a map: |
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44 | map phi=basering,i1 such that |
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45 | k[i2] --> k[var(1),...,var(n)]/phi(id) |
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46 | is a Noether normalisation |
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47 | " |
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48 | { |
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49 | def r=basering; |
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50 | int n=nvars(r); |
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51 | //----- change to lexicographical ordering ------------ |
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52 | //a procedure from ring.lib changing the order to lp |
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53 | //creating a new basering s |
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54 | def s=changeord("lp"); |
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55 | setring s; |
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56 | //----- make a random coordinate change ---------------- |
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57 | //creating lower triangular random generators for the |
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58 | //maximal ideal a procedure form random.lib |
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59 | ideal m= |
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60 | ideal(sparsetriag(n,n,0,100)*transpose(maxideal(1))); |
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61 | |
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62 | map phi=r,m; |
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63 | ideal i=std(phi(id)); |
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64 | //---------- check finiteness --------------------------- |
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65 | //from theoretical point of view Noether normalisation |
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66 | //should be o.k. but we need a test whether the |
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67 | //coordinate change was random enough |
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68 | list l=finitenessTest(i); |
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69 | |
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70 | setring r; |
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71 | list l=imap(s,l); |
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72 | |
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73 | if(l[1]==1) |
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74 | { |
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75 | //the good case, coordinate change was random enough |
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76 | return(list(fetch(s,m),l[2])); |
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77 | } |
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78 | kill s; |
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79 | //-------- the bad case, try again --------------------- |
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80 | return(noetherNormal(i)); |
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81 | } |
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82 | |
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83 | LIB"ring.lib"; |
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84 | LIB"random.lib"; |
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85 | ring R=0,(x,y,z),dp; |
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86 | ideal I = xy,xz; |
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87 | noetherNormal(I); |
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88 | |
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89 | tst_status(1);$ |
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