1 | ///////////////////////////////////////////////////////////////////////////// |
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2 | version="version ring.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: ring.lib Manipulating Rings and Maps |
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6 | |
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7 | PROCEDURES: |
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8 | changechar(c[,r]); make a copy of basering [ring r] with new char c |
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9 | changeord(o[,r]); make a copy of basering [ring r] with new ord o |
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10 | changevar(v[,r]); make a copy of basering [ring r] with new vars v |
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11 | defring(\"R\",c,n,v,o); define a ring R in specified char c, n vars v, ord o |
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12 | defrings(n[,p]); define ring Sn in n vars, char 32003 [p], ord ds |
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13 | defringp(n[,p]); define ring Pn in n vars, char 32003 [p], ord dp |
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14 | extendring(\"R\",n,v,o); extend given ring by n vars v, ord o and name it R |
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15 | fetchall(R[,str]); fetch all objects of ring R to basering |
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16 | imapall(R[,str]); imap all objects of ring R to basering |
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17 | mapall(R,i[,str]); map all objects of ring R via ideal i to basering |
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18 | ord_test(R); test wether ordering of R is global, local or mixed |
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19 | ringtensor(s,t,..); create ring, tensor product of rings s,t,... |
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20 | ringweights(r); intvec of weights of ring variables of ring r |
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21 | preimageLoc(R,phi,Q) computes preimage for non-global orderings |
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22 | rootofUnity(n); the minimal polynomial for the n-th primitive root of unity |
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23 | (parameters in square brackets [] are optional) |
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24 | "; |
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25 | |
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26 | LIB "inout.lib"; |
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27 | LIB "general.lib"; |
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28 | LIB "primdec.lib"; |
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29 | /////////////////////////////////////////////////////////////////////////////// |
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30 | |
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31 | proc changechar (list @L, list #) |
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32 | "USAGE: changechar(c[,r]); c=list, r=ring |
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33 | RETURN: ring R, obtained from the ring r [default: r=basering], by changing |
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34 | ringlist(r)[1] to c. |
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35 | EXAMPLE: example changechar; shows an example |
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36 | " |
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37 | { |
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38 | def save_ring=basering; |
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39 | if( size(#)==0 ) { def @r=basering; } |
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40 | if(( size(#)==1 ) and (typeof(#[1])=="ring")) { def @r=#[1]; } |
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41 | setring @r; |
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42 | list rl=ringlist(@r); |
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43 | if(defined(@L)!=voice) { def @L=fetch(save_ring,@L); } |
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44 | if (size(@L)==1) { rl[1]=@L[1];} else { rl[1]=@L;} |
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45 | def Rnew=ring(rl); |
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46 | setring save_ring; |
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47 | return(Rnew); |
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48 | } |
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49 | example |
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50 | { "EXAMPLE:"; echo = 2; |
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51 | ring rr=2,A,dp; |
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52 | ring r=0,(x,y,u,v),(dp(2),ds); |
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53 | def R=changechar(ringlist(rr)); R;""; |
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54 | def R1=changechar(32003,R); setring R1; R1; |
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55 | kill R,R1; |
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56 | } |
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57 | /////////////////////////////////////////////////////////////////////////////// |
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58 | |
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59 | proc changeord (list @o, list #) |
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60 | "USAGE: changeord(neword[,r]); newordstr=list, r=ring/qring |
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61 | RETURN: ring R, obtained from the ring r [default: r=basering], by changing |
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62 | order(r) to neword. |
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63 | If, say, neword=list(list(\"wp\",intvec(2,3)),list(list(\"dp\",1:(n-2)))); |
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64 | and if the ring r exists and has n variables, the ring R will be |
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65 | equipped with the monomial ordering wp(2,3),dp. |
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66 | EXAMPLE: example changeord; shows an example |
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67 | " |
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68 | { |
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69 | def save_ring=basering; |
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70 | if( size(#)==0 ) { def @r=basering; } |
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71 | if( size(#)==1 ) { def @r=#[1]; } |
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72 | setring @r; |
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73 | list rl=ringlist(@r); |
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74 | rl[3]=@o; |
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75 | def Rnew=ring(rl); |
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76 | setring save_ring; |
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77 | return(Rnew); |
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78 | } |
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79 | example |
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80 | { "EXAMPLE:"; echo = 2; |
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81 | ring r=0,(x,y,u,v),(dp(2),ds); |
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82 | def R=changeord(list(list("wp",intvec(2,3)),list("dp",1:2))); R; ""; |
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83 | ideal i = x^2,y^2-u^3,v; |
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84 | qring Q = std(i); |
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85 | def Q'=changeord(list(list("lp",nvars(Q))),Q); setring Q'; Q'; |
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86 | kill R,Q,Q'; |
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87 | } |
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88 | /////////////////////////////////////////////////////////////////////////////// |
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89 | |
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90 | proc changevar (string vars, list #) |
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91 | "USAGE: changevar(vars[,r]); vars=string, r=ring/qring |
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92 | RETURN: ring R, obtained from the ring r [default: r=basering], by changing |
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93 | varstr(r) according to the value of vars. |
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94 | If, say, vars = \"t()\" and the ring r exists and has n |
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95 | variables, the new basering will have name R and variables |
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96 | t(1),...,t(n). |
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97 | If vars = \"a,b,c,d\", the new ring will have the variables a,b,c,d. |
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98 | NOTE: This procedure is useful in connection with the procedure ringtensor, |
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99 | when a conflict between variable names must be avoided. |
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100 | This proc uses 'execute' or calls a procedure using 'execute'. |
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101 | EXAMPLE: example changevar; shows an example |
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102 | " |
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103 | { |
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104 | if( size(#)==0 ) { def @r=basering; } |
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105 | if( size(#)==1 ) { def @r=#[1]; } |
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106 | setring @r; |
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107 | ideal i = ideal(@r); int @q = size(i); |
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108 | if( @q!=0 ) |
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109 | { string @s = "Rnew1"; } |
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110 | else |
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111 | { string @s = "Rnew"; } |
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112 | string @newring = @s+"=("+charstr(@r)+"),("; |
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113 | if( vars[size(vars)-1]=="(" and vars[size(vars)]==")" ) |
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114 | { |
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115 | @newring = @newring+vars[1,size(vars)-2]+"(1.."+string(nvars(@r))+")"; |
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116 | } |
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117 | else { @newring = @newring+vars; } |
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118 | @newring = @newring+"),("+ordstr(@r)+");"; |
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119 | execute("ring "+@newring); |
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120 | if( @q!=0 ) |
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121 | { |
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122 | map phi = @r,maxideal(1); |
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123 | ideal i = phi(i); |
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124 | attrib(i,"isSB",1); //*** attrib funktioniert ? |
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125 | qring Rnew=i; |
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126 | } |
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127 | return(Rnew); |
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128 | } |
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129 | example |
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130 | { "EXAMPLE:"; echo = 2; |
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131 | ring r=0,(x,y,u,v),(dp(2),ds); |
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132 | ideal i = x^2,y^2-u^3,v; |
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133 | qring Q = std(i); |
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134 | setring(r); |
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135 | def R=changevar("A()"); R; ""; |
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136 | def Q'=changevar("a,b,c,d",Q); setring Q'; Q'; |
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137 | kill R,Q,Q'; |
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138 | } |
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139 | /////////////////////////////////////////////////////////////////////////////// |
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140 | |
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141 | proc defring (string s2, int n, string s3, string s4) |
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142 | "USAGE: defring(ch,n,va,or); ch,va,or=strings, n=integer |
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143 | RETURN: ring R with characteristic 'ch', ordering 'or' and n variables with |
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144 | names derived from va. |
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145 | If va is a single letter, say va=\"a\", and if n<=26 then a and the |
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146 | following n-1 letters from the alphabet (cyclic order) are taken as |
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147 | variables. If n>26 or if va is a single letter followed by a bracket, |
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148 | say va=\"T(\", the variables are T(1),...,T(n). |
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149 | NOTE: This proc is useful for defining a ring in a procedure. |
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150 | This proc uses 'execute' or calls a procedure using 'execute'. |
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151 | EXAMPLE: example defring; shows an example |
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152 | " |
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153 | { |
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154 | string @newring = "ring newring =("+s2+"),("; |
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155 | if( n>26 or s3[2]=="(" ) { string @v = s3[1]+"(1.."+string(n)+")"; } |
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156 | else { string @v = A_Z(s3,n); } |
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157 | @newring=@newring+@v+"),("+s4+");"; |
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158 | execute(@newring); |
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159 | return(newring); |
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160 | } |
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161 | example |
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162 | { "EXAMPLE:"; echo = 2; |
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163 | def r=defring("0",5,"u","ls"); r; setring r;""; |
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164 | def R=defring("2,A",10,"x(","dp(3),ws(1,2,3),ds"); R; setring R; |
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165 | kill R,r; |
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166 | } |
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167 | /////////////////////////////////////////////////////////////////////////////// |
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168 | |
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169 | proc defrings (int n, list #) |
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170 | "USAGE: defrings(n,[p]); n,p integers |
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171 | RETURN: ring R with characteristic p [default: p=32003], ordering ds and n |
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172 | variables x,y,z,a,b,...if n<=26 (resp. x(1..n) if n>26) |
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173 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
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174 | EXAMPLE: example defrings; shows an example |
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175 | " |
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176 | { |
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177 | int p; |
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178 | if (size(#)==0) { p=32003; } |
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179 | else { p=#[1]; } |
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180 | if (n >26) |
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181 | { |
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182 | string s="ring S ="+string(p)+",x(1.."+string(n)+"),ds;"; |
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183 | } |
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184 | else |
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185 | { |
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186 | string s="ring S ="+string(p)+",("+A_Z("x",n)+"),ds;"; |
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187 | } |
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188 | execute(s); |
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189 | dbprint(printlevel-voice+2," |
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190 | // 'defrings' created a ring. To see the ring, type (if the name R was |
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191 | // assigned to the return value): |
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192 | show R; |
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193 | // To make the ring the active basering, type |
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194 | setring R; "); |
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195 | return(S); |
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196 | } |
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197 | example |
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198 | { "EXAMPLE:"; echo = 2; |
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199 | def S5=defrings(5,0); S5; ""; |
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200 | def S30=defrings(30); S30; |
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201 | kill S5,S30; |
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202 | } |
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203 | /////////////////////////////////////////////////////////////////////////////// |
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204 | |
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205 | proc defringp (int n,list #) |
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206 | "USAGE: defringp(n,[p]); n,p=integers |
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207 | RETURN: ring R with characteristic p [default: p=32003], ordering dp and n |
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208 | variables x,y,z,a,b,...if n<=26 (resp. x(1..n) if n>26) |
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209 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
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210 | EXAMPLE: example defringp; shows an example |
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211 | " |
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212 | { |
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213 | int p; |
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214 | if (size(#)==0) { p=32003; } |
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215 | else { p=#[1]; } |
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216 | if (n >26) |
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217 | { |
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218 | string s="ring P="+string(p)+",x(1.."+string(n)+"),dp;"; |
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219 | } |
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220 | else |
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221 | { |
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222 | string s="ring P="+string(p)+",("+A_Z("x",n)+"),dp;"; |
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223 | } |
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224 | execute(s); |
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225 | dbprint(printlevel-voice+2," |
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226 | // 'defringp' created a ring. To see the ring, type (if the name R was |
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227 | // assigned to the return value): |
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228 | show R; |
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229 | // To make the ring the active basering, type |
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230 | setring R; "); |
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231 | return(P); |
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232 | } |
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233 | example |
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234 | { "EXAMPLE:"; echo = 2; |
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235 | def P5=defringp(5,0); P5; ""; |
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236 | def P30=defringp(30); P30; |
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237 | kill P5,P30; |
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238 | } |
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239 | /////////////////////////////////////////////////////////////////////////////// |
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240 | |
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241 | proc extendring (int n, string va, string o, list #) |
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242 | "USAGE: extendring(n,va,o[,iv,i,r]); va,o=strings, n,i=integers, r=ring, |
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243 | iv=intvec of positive integers or iv=0 |
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244 | RETURN: ring R, which extends the ring r by adding n new variables in front |
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245 | of (resp. after, if i!=0) the old variables. |
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246 | [default: (i,r)=(0,basering)]. |
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247 | @* -- The characteristic is the characteristic of r. |
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248 | @* -- The new vars are derived from va. If va is a single letter, say |
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249 | va=\"T\", and if n<=26 then T and the following n-1 letters from |
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250 | T..Z..T (resp. T(1..n) if n>26) are taken as additional variables. |
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251 | If va is a single letter followed by a bracket, say va=\"x(\", |
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252 | the new variables are x(1),...,x(n). |
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253 | @* -- The ordering is the product ordering of the ordering of r and of an |
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254 | ordering derived from `o` [and iv]. |
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255 | @* - If o contains a 'c' or a 'C' in front resp. at the end, this is |
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256 | taken for the whole ordering in front, resp. at the end. If o does |
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257 | not contain a 'c' or a 'C' the same rule applies to ordstr(r). |
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258 | @* - If no intvec iv is given, or if iv=0, o may be any allowed ordstr, |
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259 | like \"ds\" or \"dp(2),wp(1,2,3),Ds(2)\" or \"ds(a),dp(b),ls\" if |
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260 | a and b are globally (!) defined integers and if a+b+1<=n. |
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261 | If, however, a and b are local to a proc calling extendring, the |
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262 | intvec iv must be used to let extendring know the values of a and b |
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263 | @* - If a non-zero intvec iv is given, iv[1],iv[2],... are taken for the |
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264 | 1st, 2nd,... block of o, if o contains no substring \"w\" or \"W\" |
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265 | i.e. no weighted ordering (in the above case o=\"ds,dp,ls\" |
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266 | and iv=a,b). |
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267 | If o contains a weighted ordering (only one (!) weighted block is |
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268 | allowed) iv[1] is taken as size for the weight-vector, the next |
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269 | iv[1] values of iv are taken as weights and the remaining values of |
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270 | iv as block size for the remaining non-weighted blocks. |
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271 | e.g. o=\"dp,ws,Dp,ds\", iv=3,2,3,4,2,5 creates the ordering |
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272 | dp(2),ws(2,3,4),Dp(5),ds |
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273 | NOTE: This proc is useful for adding deformation parameters. |
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274 | This proc uses 'execute' or calls a procedure using 'execute'. |
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275 | If you use it in your own proc, it may be advisable to let the local |
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276 | names of your proc start with a @ |
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277 | EXAMPLE: example extendring; shows an example |
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278 | " |
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279 | { |
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280 | //--------------- initialization and place c/C of ordering properly ----------- |
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281 | string @o1,@o2,@ro,@wstr,@v,@newring; |
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282 | int @i,@w,@ii,@k; |
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283 | intvec @iv,@iw; |
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284 | if( find(o,"c")+find(o,"C") != 0) |
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285 | { |
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286 | @k=1; |
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287 | if( o[1]=="c" or o[1]=="C" ) { @o1=o[1,2]; o=o[3..size(o)]; } |
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288 | else { @o2=o[size(o)-1,2]; o=o[1..size(o)-2]; } |
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289 | } |
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290 | if( size(#)==0 ) { #[1]=0; } |
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291 | if( typeof(#[1])!="intvec" ) |
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292 | { |
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293 | if( size(#)==1 ) { @i=#[1]; def @r=basering; } |
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294 | if( size(#)==2 ) { @i=#[1]; def @r=#[2]; } |
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295 | if( o[size(o)]!=")" and find(o,",")==0 ) { o=o+"("+string(n)+")"; } |
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296 | } |
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297 | else |
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298 | { |
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299 | @iv=#[1]; |
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300 | if( size(#)==2 ) { @i=#[2]; def @r=basering; } |
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301 | if( size(#)==3 ) { @i=#[2]; def @r=#[3]; } |
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302 | if( @iv==0 && o[size(o)]!=")" && find(o,",")==0 ) {o=o+"("+string(n)+")";} |
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303 | } |
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304 | @ro=ordstr(@r); |
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305 | if( @ro[1]=="c" or @ro[1]=="C" ) |
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306 | { @v=@ro[1,2]; @ro=@ro[3..size(@ro)]; } |
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307 | else |
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308 | { @wstr=@ro[size(@ro)-1,2]; @ro=@ro[1..size(@ro)-2]; } |
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309 | if( @k==0) { @o1=@v; @o2=@wstr; } |
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310 | //----------------- prepare ordering if an intvec is given -------------------- |
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311 | if( typeof(#[1])=="intvec" and #[1]!=0 ) |
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312 | { |
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313 | @k=n; //@k counts no of vars not yet ordered |
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314 | @w=find(o,"w")+find(o,"W");o=o+" "; |
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315 | if( @w!=0 ) |
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316 | { |
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317 | @wstr=o[@w..@w+1]; |
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318 | o=o[1,@w-1]+"@"+o[@w+2,size(o)]; |
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319 | @iw=@iv[2..@iv[1]+1]; |
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320 | @wstr=@wstr+"("+string(@iw)+")"; |
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321 | @k=@k-@iv[1]; |
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322 | @iv=@iv[@iv[1]+2..size(@iv)]; |
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323 | @w=0; |
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324 | } |
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325 | for( @ii=1; @ii<=size(@iv); @ii=@ii+1 ) |
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326 | { |
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327 | if( find(o,",",@w+1)!=0 ) |
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328 | { |
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329 | @w=find(o,",",@w+1); |
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330 | if( o[@w-1]!="@" ) |
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331 | { |
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332 | o=o[1,@w-1]+"("+string(@iv[@ii])+")"+o[@w,size(o)]; |
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333 | @w=find(o,",",@w+1); |
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334 | @k=@k-@iv[@ii]; |
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335 | } |
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336 | else { @ii=@ii-1; } |
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337 | } |
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338 | } |
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339 | @w=find(o,"@"); |
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340 | if( @w!=0 ) { o=o[1,@w-1] + @wstr + o[@w+1,size(o)]; } |
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341 | if( @k>0 and o[size(o)]!=")" ) { o=o+"("+string(@k)+")"; } |
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342 | } |
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343 | //------------------------ prepare string of new ring ------------------------- |
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344 | @newring = "ring na =("+charstr(@r)+"),("; |
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345 | if( n>26 or va[2]=="(" ) { @v = va[1]+"(1.."+string(n)+")"; } |
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346 | else { @v = A_Z(va,n); } |
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347 | if( @i==0 ) |
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348 | { |
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349 | @v=@v+","+varstr(@r); |
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350 | o=@o1+o+","+@ro+@o2; |
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351 | } |
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352 | else |
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353 | { |
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354 | @v=varstr(@r)+","+@v; |
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355 | o=@o1+@ro+","+o+@o2; |
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356 | } |
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357 | @newring=@newring+@v+"),("+o+");"; |
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358 | //---------------------------- execute and export ----------------------------- |
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359 | execute(@newring); |
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360 | dbprint(printlevel-voice+2," |
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361 | // 'extendring' created a new ring. |
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362 | // To see the ring, type (if the name 'R' was assigned to the return value): |
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363 | show(R); |
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364 | "); |
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365 | |
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366 | return(na); |
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367 | } |
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368 | example |
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369 | { "EXAMPLE:"; echo = 2; |
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370 | ring r=0,(x,y,z),ds; |
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371 | show(r);""; |
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372 | // blocksize is derived from no of vars: |
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373 | int t=5; |
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374 | def R1=extendring(t,"a","dp"); //t global: "dp" -> "dp(5)" |
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375 | show(R1); setring R1; ""; |
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376 | def R2=extendring(4,"T(","c,dp",1,r); //"dp" -> "c,..,dp(4)" |
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377 | show(R2); setring R2; ""; |
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378 | |
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379 | // no intvec given, blocksize given: given blocksize is used: |
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380 | def R3=extendring(4,"T(","dp(2)",0,r); // "dp(2)" -> "dp(2)" |
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381 | show(R3); setring R3; ""; |
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382 | |
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383 | // intvec given: weights and blocksize is derived from given intvec |
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384 | // (no specification of a blocksize in the given ordstr is allowed!) |
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385 | // if intvec does not cover all given blocks, the last block is used |
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386 | // for the remaining variables, if intvec has too many components, |
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387 | // the last ones are ignored |
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388 | intvec v=3,2,3,4,1,3; |
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389 | def R4=extendring(10,"A","ds,ws,Dp,dp",v,0,r); |
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390 | // v covers 3 blocks: v[1] (=3) : no of components of ws |
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391 | // next v[1] values (=v[2..4]) give weights |
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392 | // remaining components of v are used for the remaining blocks |
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393 | show(R4); |
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394 | kill r,R1,R2,R3,R4; |
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395 | } |
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396 | /////////////////////////////////////////////////////////////////////////////// |
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397 | |
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398 | proc fetchall (def R, list #) |
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399 | "USAGE: fetchall(R[,s]); R=ring/qring, s=string |
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400 | CREATE: fetch all objects of ring R (of type poly/ideal/vector/module/number/matrix) |
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401 | into the basering. |
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402 | If no 2nd argument is present, the names are the same as in R. If, |
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403 | say, f is a polynomial in R and the 2nd argument is the string \"R\", then f |
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404 | is mapped to f_R etc. |
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405 | RETURN: no return value |
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406 | NOTE: As fetch, this procedure maps the 1st, 2nd, ... variable of R to the |
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407 | 1st, 2nd, ... variable of the basering. |
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408 | The 2nd argument is useful in order to avoid conflicts of names, the |
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409 | empty string is allowed |
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410 | CAUTION: fetchall does not work for locally defined names. |
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411 | It does not work if R contains a map. |
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412 | SEE ALSO: imapall |
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413 | EXAMPLE: example fetchall; shows an example |
---|
414 | " |
---|
415 | { |
---|
416 | list @L@=names(R); |
---|
417 | int @ii@; string @s@; |
---|
418 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
419 | for( @ii@=size(@L@); @ii@>0; @ii@-- ) |
---|
420 | { |
---|
421 | execute("def "+@L@[@ii@]+@s@+"=fetch(R,`@L@[@ii@]`);"); |
---|
422 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
423 | } |
---|
424 | return(); |
---|
425 | } |
---|
426 | example |
---|
427 | { "EXAMPLE:"; echo=2; |
---|
428 | // The example is not shown since fetchall does not work in a procedure; |
---|
429 | // (and hence not in the example procedure). Try the following commands: |
---|
430 | // ring R=0,(x,y,z),dp; |
---|
431 | // ideal j=x,y2,z2; |
---|
432 | // matrix M[2][3]=1,2,3,x,y,z; |
---|
433 | // j; print(M); |
---|
434 | // ring S=0,(a,b,c),ds; |
---|
435 | // fetchall(R); //map from R to S: x->a, y->b, z->c; |
---|
436 | // names(S); |
---|
437 | // j; print(M); |
---|
438 | // fetchall(S,"1"); //identity map of S: copy objects, change names |
---|
439 | // names(S); |
---|
440 | // kill R,S; |
---|
441 | } |
---|
442 | /////////////////////////////////////////////////////////////////////////////// |
---|
443 | |
---|
444 | proc imapall (def R, list #) |
---|
445 | "USAGE: imapall(R[,s]); R=ring/qring, s=string |
---|
446 | CREATE: map all objects of ring R (of type poly/ideal/vector/module/number/matrix) |
---|
447 | into the basering by applying imap to all objects of R. |
---|
448 | If no 2nd argument is present, the names are the same as in R. If, |
---|
449 | say, f is a polynomial in R and the 3rd argument is the string \"R\", then f |
---|
450 | is mapped to f_R etc. |
---|
451 | RETURN: no return value |
---|
452 | NOTE: As imap, this procedure maps the variables of R to the variables with |
---|
453 | the same name in the basering, the other variables are mapped to 0. |
---|
454 | The 2nd argument is useful in order to avoid conflicts of names, the |
---|
455 | empty string is allowed |
---|
456 | CAUTION: imapall does not work for locally defined names. |
---|
457 | It does not work if R contains a map |
---|
458 | SEE ALSO: fetchall |
---|
459 | EXAMPLE: example imapall; shows an example |
---|
460 | " |
---|
461 | { |
---|
462 | list @L@=names(R); |
---|
463 | int @ii@; string @s@; |
---|
464 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
465 | for( @ii@=size(@L@); @ii@>0; @ii@-- ) |
---|
466 | { |
---|
467 | execute("def "+@L@[@ii@]+@s@+"=imap(R,`@L@[@ii@]`);"); |
---|
468 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
469 | } |
---|
470 | return(); |
---|
471 | } |
---|
472 | example |
---|
473 | { "EXAMPLE:"; echo = 2; |
---|
474 | // The example is not shown since imapall does not work in a procedure |
---|
475 | // (and hence not in the example procedure). Try the following commands: |
---|
476 | // ring R=0,(x,y,z,u),dp; |
---|
477 | // ideal j=x,y,z,u2+ux+z; |
---|
478 | // matrix M[2][3]=1,2,3,x,y,uz; |
---|
479 | // j; print(M); |
---|
480 | // ring S=0,(a,b,c,x,z,y),ds; |
---|
481 | // imapall(R); //map from R to S: x->x, y->y, z->z, u->0 |
---|
482 | // names(S); |
---|
483 | // j; print(M); |
---|
484 | // imapall(S,"1"); //identity map of S: copy objects, change names |
---|
485 | // names(S); |
---|
486 | // kill R,S; |
---|
487 | } |
---|
488 | /////////////////////////////////////////////////////////////////////////////// |
---|
489 | |
---|
490 | proc mapall (def R, ideal i, list #) |
---|
491 | "USAGE: mapall(R,i[,s]); R=ring/qring, i=ideal of basering, s=string |
---|
492 | CREATE: map all objects of ring R (of type poly/ideal/vector/module/number/ |
---|
493 | matrix, map) into the basering by mapping the j-th variable of R to |
---|
494 | the j-th generator of the ideal i. If no 3rd argument is present, the |
---|
495 | names are the same as in R. If, say, f is a polynomial in R and the 3rd |
---|
496 | argument is the string \"R\", then f is mapped to f_R etc. |
---|
497 | RETURN: no return value. |
---|
498 | NOTE: This procedure has the same effect as defining a map, say psi, by |
---|
499 | map psi=R,i; and then applying psi to all objects of R. In particular, |
---|
500 | maps from R to some ring S are composed with psi, creating thus a map |
---|
501 | from the basering to S. |
---|
502 | mapall may be combined with copyring to change vars for all objects. |
---|
503 | The 3rd argument is useful in order to avoid conflicts of names, the |
---|
504 | empty string is allowed. |
---|
505 | CAUTION: mapall does not work for locally defined names. |
---|
506 | EXAMPLE: example mapall; shows an example |
---|
507 | " |
---|
508 | { |
---|
509 | list @L@=names(R); map @psi@ = R,i; |
---|
510 | int @ii@; string @s@; |
---|
511 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
512 | for( @ii@=size(@L@); @ii@>0; @ii@-- ) |
---|
513 | { |
---|
514 | execute("def "+@L@[@ii@]+@s@+"=@psi@(`@L@[@ii@]`);"); |
---|
515 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
516 | } |
---|
517 | return(); |
---|
518 | } |
---|
519 | example |
---|
520 | { "EXAMPLE:"; echo = 2; |
---|
521 | // The example is not shown since mapall does not work in a procedure |
---|
522 | // (and hence not in the example procedure). Try the following commands: |
---|
523 | // ring R=0,(x,y,z),dp; |
---|
524 | // ideal j=x,y,z; |
---|
525 | // matrix M[2][3]=1,2,3,x,y,z; |
---|
526 | // map phi=R,x2,y2,z2; |
---|
527 | // ring S=0,(a,b,c),ds; |
---|
528 | // ideal i=c,a,b; |
---|
529 | // mapall(R,i); //map from R to S: x->c, y->a, z->b |
---|
530 | // names(S); |
---|
531 | // j; print(M); phi; //phi maps R to S: x->c2, y->a2, z->b2 |
---|
532 | // ideal i1=a2,a+b,1; |
---|
533 | // mapall(R,i1,\"\"); //map from R to S: x->a2, y->a+b, z->1 |
---|
534 | // names(S); |
---|
535 | // j_; print(M_); phi_; |
---|
536 | // changevar(\"T\",\"x()\",R); //change vars in R and call result T |
---|
537 | // mapall(R,maxideal(1)); //identity map from R to T |
---|
538 | // names(T); |
---|
539 | // j; print(M); phi; |
---|
540 | // kill R,S,T; |
---|
541 | } |
---|
542 | /////////////////////////////////////////////////////////////////////////////// |
---|
543 | |
---|
544 | proc ord_test (def r) |
---|
545 | "USAGE: ord_test(r); r ring/qring |
---|
546 | RETURN: int 1 (resp. -1, resp. 0) if ordering of r is global (resp. local, |
---|
547 | resp. mixed) |
---|
548 | SEE ALSO: attrib |
---|
549 | EXAMPLE: example ord_test; shows an example |
---|
550 | " |
---|
551 | { |
---|
552 | if ((typeof(r) != "ring") and (typeof(r) != "qring")) |
---|
553 | { |
---|
554 | ERROR("ord_test requires a ring/qring as input"); |
---|
555 | } |
---|
556 | if (attrib(r,"global")==1) { return(1);} |
---|
557 | def BAS = basering; |
---|
558 | setring r; |
---|
559 | poly f; |
---|
560 | int n,o,u = nvars(r),1,1; |
---|
561 | int ii; |
---|
562 | for ( ii=1; ii<=n; ii++ ) |
---|
563 | { |
---|
564 | f = 1+var(ii); |
---|
565 | o = o*(lead(f) == var(ii)); |
---|
566 | u = u*(lead(f) == 1); |
---|
567 | } |
---|
568 | setring BAS; |
---|
569 | if ( o==1 ) { return(1); } |
---|
570 | if ( u==1 ) { return(-1); } |
---|
571 | else { return(0); } |
---|
572 | } |
---|
573 | example |
---|
574 | { "EXAMPLE:"; echo = 2; |
---|
575 | ring R = 0,(x,y),dp; |
---|
576 | ring S = 0,(u,v),ls; |
---|
577 | ord_test(R); |
---|
578 | ord_test(S); |
---|
579 | ord_test(R+S); |
---|
580 | } |
---|
581 | /////////////////////////////////////////////////////////////////////////////// |
---|
582 | |
---|
583 | proc ringtensor (list #) |
---|
584 | "USAGE: ringtensor(r1,r2,...); r1,r2,...=rings |
---|
585 | RETURN: ring R whose variables are the variables from all rings r1,r2,... |
---|
586 | and whose monomial ordering is the block (product) ordering of the |
---|
587 | respective monomial orderings of r1,r2,... . Hence, R |
---|
588 | is the tensor product of the rings r1,r2,... with ordering matrix |
---|
589 | equal to the direct sum of the ordering matrices of r1,r2,... |
---|
590 | NOTE: The characteristic of the new ring will be p if one ring has |
---|
591 | characteristic p. The names of variables in the rings r1,r2,... |
---|
592 | must differ. |
---|
593 | The procedure works also for quotient rings ri, if the characteristic |
---|
594 | of ri is compatible with the characteristic of the result |
---|
595 | (i.e. if imap from ri to the result is implemented) |
---|
596 | SEE ALSO: ring operations |
---|
597 | EXAMPLE: example ringtensor; shows an example |
---|
598 | " |
---|
599 | { |
---|
600 | int @i; |
---|
601 | int @n = size(#); |
---|
602 | if (@n<=1) { ERROR("at least 2 rings required"); } |
---|
603 | def @s=#[1]+#[2]; |
---|
604 | for (@i=3; @i<=@n;@i++) |
---|
605 | { |
---|
606 | def @ss=@s+#[@i]; |
---|
607 | kill @s; |
---|
608 | def @s=@ss; |
---|
609 | kill @ss; |
---|
610 | } |
---|
611 | dbprint(printlevel-voice+2," |
---|
612 | // 'ringtensor' created a ring. To see the ring, type (if the name R was |
---|
613 | // assigned to the return value): |
---|
614 | show(R); |
---|
615 | // To make the ring the active basering, type |
---|
616 | setring R; "); |
---|
617 | return(@s); |
---|
618 | } |
---|
619 | example |
---|
620 | { "EXAMPLE:"; echo = 2; |
---|
621 | ring r=32003,(x,y,u,v),dp; |
---|
622 | ring s=0,(a,b,c),wp(1,2,3); |
---|
623 | ring t=0,x(1..5),(c,ls); |
---|
624 | def R=ringtensor(r,s,t); |
---|
625 | type R; |
---|
626 | setring s; |
---|
627 | ideal i = a2+b3+c5; |
---|
628 | def S=changevar("x,y,z"); //change vars of s |
---|
629 | setring S; |
---|
630 | qring qS =std(fetch(s,i)); //create qring of S mod i (mapped to S) |
---|
631 | def T=changevar("d,e,f,g,h",t); //change vars of t |
---|
632 | setring T; |
---|
633 | qring qT=std(d2+e2-f3); //create qring of T mod d2+e2-f3 |
---|
634 | def Q=ringtensor(s,qS,t,qT); |
---|
635 | setring Q; type Q; |
---|
636 | kill R,S,T,Q; |
---|
637 | } |
---|
638 | /////////////////////////////////////////////////////////////////////////////// |
---|
639 | |
---|
640 | proc ringweights (def P) |
---|
641 | "USAGE: ringweights(P); P=name of an existing ring (true name, not a string) |
---|
642 | RETURN: intvec consisting of the weights of the variables of P, as they |
---|
643 | appear when typing P;. |
---|
644 | NOTE: This is useful when enlarging P but keeping the weights of the old |
---|
645 | variables. |
---|
646 | EXAMPLE: example ringweights; shows an example |
---|
647 | " |
---|
648 | { |
---|
649 | int i; |
---|
650 | intvec rw; |
---|
651 | //------------------------- find weights ------------------------- |
---|
652 | for(i=nvars(P);i>0;i--) |
---|
653 | { rw[i]=ord(var(i)); } |
---|
654 | return(rw); |
---|
655 | } |
---|
656 | example |
---|
657 | {"EXAMPLE:"; echo = 2; |
---|
658 | ring r0 = 0,(x,y,z),dp; |
---|
659 | ringweights(r0); |
---|
660 | ring r1 = 0,x(1..5),(ds(3),wp(2,3)); |
---|
661 | ringweights(r1);""; |
---|
662 | // an example for enlarging the ring, keeping the first weights: |
---|
663 | intvec v = ringweights(r1),6,2,3,4,5; |
---|
664 | ring R = 0,x(1..10),(a(v),dp); |
---|
665 | ordstr(R); |
---|
666 | } |
---|
667 | /////////////////////////////////////////////////////////////////////////////// |
---|
668 | proc preimageLoc(string R_name,string phi_name,string Q_name ) |
---|
669 | "USAGE: preimageLoc ( ring_name, map_name, ideal_name ); |
---|
670 | all input parameters of type string |
---|
671 | RETURN: ideal |
---|
672 | PURPOSE: compute the preimage of an ideal under a given map for non-global |
---|
673 | orderings. |
---|
674 | The 2nd argument has to be the name of a map from the basering to |
---|
675 | the given ring (or the name of an ideal defining such a map), and |
---|
676 | the ideal has to be an ideal in the given ring. |
---|
677 | SEE ALSO: preimage |
---|
678 | KEYWORDS: preimage under a map between local rings, map between local rings, map between local and global rings |
---|
679 | EXAMPLE: example preimageLoc ; shows an example |
---|
680 | "{ |
---|
681 | def S=basering; |
---|
682 | int i; |
---|
683 | string newRing,minpoly_string; |
---|
684 | if(attrib(S,"global")!=1) |
---|
685 | { |
---|
686 | if(typeof(S)=="qring") |
---|
687 | { |
---|
688 | ideal I=ideal(S); |
---|
689 | newRing="ring S0=("+charstr(S)+"),("+varstr(S)+"),dp;"; |
---|
690 | minpoly_string=string(minpoly); |
---|
691 | execute(newRing); |
---|
692 | execute("minpoly="+minpoly_string+";"); |
---|
693 | ideal I=imap(S,I); |
---|
694 | list pr=primdecGTZ(I); |
---|
695 | newRing="ring SL=("+charstr(S)+"),("+varstr(S)+"),("+ordstr(S)+");"; |
---|
696 | execute(newRing); |
---|
697 | execute("minpoly="+minpoly_string+";"); |
---|
698 | list pr=imap(S0,pr); |
---|
699 | ideal I0=std(pr[1][1]); |
---|
700 | for(i=2;i<=size(pr);i++) |
---|
701 | { |
---|
702 | I0=intersect(I0,std(pr[i][1])); |
---|
703 | } |
---|
704 | setring S0; |
---|
705 | ideal I0=imap(SL,I0); |
---|
706 | qring S1=std(I0); |
---|
707 | } |
---|
708 | else |
---|
709 | { |
---|
710 | def S1=S; |
---|
711 | } |
---|
712 | } |
---|
713 | else |
---|
714 | { |
---|
715 | def S1=S; |
---|
716 | } |
---|
717 | def @R=`R_name`; |
---|
718 | setring @R; |
---|
719 | def @phi=`phi_name`; |
---|
720 | ideal phiId=ideal(@phi); |
---|
721 | def Q=`Q_name`; |
---|
722 | if(attrib(@R,"global")!=1) |
---|
723 | { |
---|
724 | if(typeof(@R)=="qring") |
---|
725 | { |
---|
726 | ideal J=ideal(@R); |
---|
727 | newRing="ring R0=("+charstr(@R)+"),("+varstr(@R)+"),dp;"; |
---|
728 | minpoly_string=string(minpoly); |
---|
729 | execute(newRing); |
---|
730 | execute("minpoly="+minpoly_string+";"); |
---|
731 | ideal J=imap(@R,J); |
---|
732 | list pr=primdecGTZ(J); |
---|
733 | newRing="ring RL=("+charstr(@R)+"),("+varstr(@R)+"),("+ordstr(@R)+");"; |
---|
734 | execute(newRing); |
---|
735 | execute("minpoly="+minpoly_string+";"); |
---|
736 | list pr=imap(R0,pr); |
---|
737 | ideal J0=std(pr[1][1]); |
---|
738 | for(i=2;i<=size(pr);i++) |
---|
739 | { |
---|
740 | J0=intersect(J0,std(pr[i][1])); |
---|
741 | } |
---|
742 | setring R0; |
---|
743 | ideal J0=imap(RL,J0); |
---|
744 | qring R1=std(J0); |
---|
745 | ideal Q=imap(@R,Q); |
---|
746 | map @phi=S1,imap(@R,phiId); |
---|
747 | } |
---|
748 | else |
---|
749 | { |
---|
750 | def R1=@R; |
---|
751 | } |
---|
752 | } |
---|
753 | else |
---|
754 | { |
---|
755 | def R1=@R; |
---|
756 | } |
---|
757 | setring S1; |
---|
758 | ideal preQ=preimage(R1,@phi,Q); |
---|
759 | setring S; |
---|
760 | ideal prQ=imap(S1,preQ); |
---|
761 | return(prQ); |
---|
762 | } |
---|
763 | example |
---|
764 | { "EXAMPLE:"; echo=2; |
---|
765 | ring S =0,(x,y,z),dp; |
---|
766 | ring R0=0,(x,y,z),ds; |
---|
767 | qring R=std(x+x2); |
---|
768 | map psi=S,x,y,z; |
---|
769 | ideal null; |
---|
770 | setring S; |
---|
771 | ideal nu=preimageLoc("R","psi","null"); |
---|
772 | nu; |
---|
773 | } |
---|
774 | |
---|
775 | ////////////////////////////////////////////////////////////////////////////// |
---|
776 | /* moved here from the nctools.lib */ |
---|
777 | /////////////////////////////////////////////////////////////////////////////// |
---|
778 | proc rootofUnity(int n) |
---|
779 | "USAGE: rootofUnity(n); n an integer |
---|
780 | RETURN: number |
---|
781 | PURPOSE: compute the minimal polynomial for the n-th primitive root of unity |
---|
782 | NOTE: works only in field extensions by one element |
---|
783 | EXAMPLE: example rootofUnity; shows examples |
---|
784 | " |
---|
785 | { |
---|
786 | if ( npars(basering) !=1 ) |
---|
787 | { |
---|
788 | "the procedure works only with one parameter"; |
---|
789 | return(0); |
---|
790 | } |
---|
791 | if (n<1) { return(0); } |
---|
792 | number mp = par(1); |
---|
793 | if (n==1) { return(mp-1); } |
---|
794 | if (n==2) { return(mp+1); } |
---|
795 | def OldRing = basering; |
---|
796 | string CH = charstr(basering); |
---|
797 | string MCH; |
---|
798 | int j=1; |
---|
799 | while ( (CH[j] !=",") && (j<=size(CH))) |
---|
800 | { |
---|
801 | MCH=MCH+CH[j]; j++; |
---|
802 | } |
---|
803 | string SR = "ring @@rR="+MCH+","+parstr(basering)+",dp;"; |
---|
804 | execute(SR); |
---|
805 | poly @t=var(1)^n-1; // (x^2i-1)=(x^i-1)(x^i+1) |
---|
806 | list l=factorize(@t); |
---|
807 | ideal @l=l[1]; |
---|
808 | list @d; |
---|
809 | int s=size(@l); |
---|
810 | int d=deg(@l[s]); |
---|
811 | int cnt=1; |
---|
812 | poly res; |
---|
813 | for (j=s-1; j>=1; j--) |
---|
814 | { |
---|
815 | if ( deg(@l[j]) > d) { d=deg(@l[j]); } |
---|
816 | } |
---|
817 | for (j=1; j<=s; j++) |
---|
818 | { |
---|
819 | if ( deg(@l[j]) == d) { @d[cnt]=@l[j]; cnt++; } |
---|
820 | } |
---|
821 | if ( size(@d)==1 ) |
---|
822 | { |
---|
823 | res = poly(@d[1]); |
---|
824 | } |
---|
825 | else |
---|
826 | { |
---|
827 | j=1; |
---|
828 | while ( j <= size(@d) ) |
---|
829 | { |
---|
830 | res = @d[j]-lead(@d[j]); |
---|
831 | if ( leadcoef(res) >=0 ) { j++; } |
---|
832 | else { break; } |
---|
833 | } |
---|
834 | res = @d[j]; |
---|
835 | } |
---|
836 | setring OldRing; |
---|
837 | poly I = imap(@@rR,res); |
---|
838 | mp = leadcoef(I); |
---|
839 | kill @@rR; |
---|
840 | return(mp); |
---|
841 | } |
---|
842 | example |
---|
843 | { |
---|
844 | "EXAMPLE:";echo=2; |
---|
845 | ring r = (0,q),(x,y,z),dp; |
---|
846 | rootofUnity(6); |
---|
847 | rootofUnity(7); |
---|
848 | minpoly = rootofUnity(8); |
---|
849 | r; |
---|
850 | } |
---|
851 | |
---|
852 | |
---|
853 | |
---|
854 | |
---|
855 | proc isQuotientRing( rng ) |
---|
856 | "USAGE: isQuotientRing ( rng ); |
---|
857 | RETURN: 1 if rng is a quotient ring, 0 otherwise. |
---|
858 | PURPOSE: check if typeof a rng "qring" |
---|
859 | KEYWORDS: qring ring ideal 'factor ring' |
---|
860 | EXAMPLE: example isQuotientRing ; shows an example |
---|
861 | " |
---|
862 | { |
---|
863 | return ( size(ideal(rng)) != 0 ); |
---|
864 | } |
---|
865 | example |
---|
866 | { |
---|
867 | ring rng = 0,x,dp; |
---|
868 | isQuotientRing(rng); //no |
---|
869 | // if a certain method does not support quotient rings, |
---|
870 | // then a parameter test could be performed: |
---|
871 | ASSUME( 0, 0==isQuotientRing(basering)); |
---|
872 | |
---|
873 | qring q= ideal(x); // constructs rng/ideal(x) |
---|
874 | isQuotientRing(q); // yes |
---|
875 | } |
---|
876 | |
---|
877 | static proc testIsQuotientRing() |
---|
878 | { |
---|
879 | ring rng = real,x,dp; |
---|
880 | ASSUME(0, 0== isQuotientRing(rng) ) ; |
---|
881 | |
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882 | qring qrng = 1; |
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883 | ASSUME(0, isQuotientRing(qrng) ) ; |
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884 | |
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885 | ring rng2 = integer,x,dp; |
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886 | ASSUME(0, 0 == isQuotientRing(rng2) ) ; |
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887 | |
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888 | qring qrng2=0; |
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889 | ASSUME(0, isQuotientRing(qrng2) ) ; |
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890 | |
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891 | ring rng3 = 0,x,dp; |
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892 | ASSUME(0, 0 == isQuotientRing(rng3) ) ; |
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893 | |
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894 | qring qrng3=1; |
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895 | ASSUME(0, isQuotientRing(qrng3) ) ; |
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896 | } |
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897 | |
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898 | |
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899 | |
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900 | |
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901 | |
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902 | proc hasIntegerCoefficientRing( rng ) |
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903 | "USAGE: hasIntegerCoefficientRing ( rng ); |
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904 | RETURN: 1 if rng is has integer ring coefficients, 0 otherwise. |
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905 | KEYWORDS: integer ring coefficients |
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906 | EXAMPLE: example hasIntegerCoefficientRing ; shows an example |
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907 | " |
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908 | proc hasIntegerCoefficientRing(rng) |
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909 | { |
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910 | def rl = ringlist(rng); |
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911 | if ( not (typeof(rl[1][1])=="string") ) { return (0); } |
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912 | return ( rl[1][1]=="integer" ); |
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913 | } |
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914 | example |
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915 | { |
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916 | ring rng = integer,x,dp; |
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917 | hasIntegerCoefficientRing(rng); //yes |
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918 | // if a certain method supports only rings with integer coefficients, |
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919 | // then a parameter test could be performed: |
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920 | ASSUME( 0, hasIntegerCoefficientRing(basering)); //ok |
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921 | |
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922 | ring rng2 = 0, x, dp; |
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923 | hasIntegerCoefficientRing(rng2); // no |
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924 | } |
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925 | |
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926 | |
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927 | static proc testHasIntegerCoefficientRing() |
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928 | { |
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929 | ring rng = integer,x,dp; |
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930 | ASSUME(0, hasIntegerCoefficientRing( rng ) ); |
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931 | |
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932 | qring q = ideal(x); |
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933 | ASSUME(0, hasIntegerCoefficientRing( q ) ); |
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934 | |
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935 | ring rng2 = 0,x,dp; |
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936 | ASSUME(0, 0==hasIntegerCoefficientRing( rng2 ) ); |
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937 | |
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938 | ring rng3 = (0,a),x,dp; |
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939 | ASSUME(0, 0==hasIntegerCoefficientRing( rng3 ) ); |
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940 | |
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941 | ring rng4 = (real,a),x,dp; |
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942 | ASSUME(0, 0==hasIntegerCoefficientRing( rng4 ) ); |
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943 | |
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944 | ring rng5 = (real),x,dp; |
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945 | ASSUME(0, 0==hasIntegerCoefficientRing( rng5 ) ); |
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946 | } |
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947 | |
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948 | |
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949 | |
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950 | |
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