1 | ///////////////////////////////////////////////////////////////////////////// |
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2 | version="version ring.lib 4.1.1.0 Dec_2017 "; // $Id$ |
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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 | AUTHORS: Singular team |
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7 | |
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8 | PROCEDURES: |
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9 | changechar(c[,r]); make a copy of basering [ring r] with new char c |
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10 | changeord(o[,r]); make a copy of basering [ring r] with new ord o |
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11 | changevar(v[,r]); make a copy of basering [ring r] with new vars v |
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12 | defring(\"R\",c,n,v,o); define a ring R in specified char c, n vars v, ord o |
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13 | defrings(n[,p]); define ring Sn in n vars, char 32003 [p], ord ds |
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14 | defringp(n[,p]); define ring Pn in n vars, char 32003 [p], ord dp |
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15 | extendring(\"R\",n,v,o); extend given ring by n vars v, ord o and name it R |
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16 | fetchall(R[,str]); fetch all objects of ring R to basering |
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17 | imapall(R[,str]); imap all objects of ring R to basering |
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18 | mapall(R,i[,str]); map all objects of ring R via ideal i to basering |
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19 | ord_test(R); test wether ordering of R is global, local or mixed |
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20 | ringtensor(s,t,..); create ring, tensor product of rings s,t,... |
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21 | ringweights(r); intvec of weights of ring variables of ring r |
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22 | preimageLoc(R,phi,Q) computes preimage for non-global orderings |
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23 | rootofUnity(n); the minimal polynomial for the n-th primitive root of unity |
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24 | (parameters in square brackets [] are optional) |
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25 | optionIsSet(opt) check if as a string given option is set or not. |
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26 | hasFieldCoefficient check if the coefficient ring is considered a field |
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27 | hasGFCoefficient check if the coefficient ring is GF(p,k) |
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28 | hasZpCoefficient check if the coefficient ring is ZZ/p |
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29 | hasZp_aCoefficient check if the coefficient ring is an elag. ext. of ZZ/p |
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30 | hasQQCoefficient check if the coefficient ring is QQ |
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31 | hasNumericCoeffs(rng) check for use of floating point numbers |
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32 | hasCommutativeVars(rng) non-commutive or commnuative polynomial ring |
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33 | hasGlobalOrdering(rng) global versus mixed/local monomial ordering |
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34 | hasMixedOrdering() mixed versus global/local ordering |
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35 | hasAlgExtensionCoefficient(r) coefficients are an algebraic extension |
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36 | hasTransExtensionCoefficient(r) coefficients are rational functions |
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37 | isQuotientRing(rng) ring is a qotient ring |
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38 | isSubModule(I,J) check if I is in J as submodule |
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39 | |
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40 | changeordTo(r,o) change the ordering of a ring to a simple one |
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41 | addvarsTo(r,vars,i) add variables to a ring |
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42 | addNvarsTo(r,N,name,i) add N variables to a ring |
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43 | "; |
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44 | |
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45 | LIB "inout.lib"; |
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46 | LIB "general.lib"; |
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47 | LIB "primdec.lib"; |
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48 | |
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49 | /////////////////////////////////////////////////////////////////////////////// |
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50 | proc optionIsSet(string optionName) |
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51 | " |
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52 | USAGE: optionIsSet( optionName ) |
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53 | PARAMETERS: optionName: a name as string of an option of interest |
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54 | RETURN: true, if the by optionName given option is active, false otherwise. |
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55 | EXAMPLE: example optionIsSet; |
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56 | " |
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57 | { |
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58 | intvec op = option(get); |
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59 | //sanity check, if option is valid. will raise an error if not |
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60 | option(optionName); option("no" + optionName); |
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61 | option(set,op); |
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62 | // first entry is currently a comment "//options:", which is not an option. |
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63 | int pos = find(option(), optionName, 11 ); |
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64 | return(pos>0); |
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65 | } |
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66 | example |
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67 | { |
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68 | // check if the option "warn" is set. |
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69 | optionIsSet("warn"); |
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70 | option("warn"); |
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71 | // now the option is set |
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72 | optionIsSet("warn"); |
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73 | |
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74 | option("nowarn"); |
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75 | // now the option is unset |
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76 | optionIsSet("warn"); |
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77 | } |
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78 | |
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79 | |
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80 | static proc testOptionIsSet() |
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81 | { |
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82 | option("warn"); |
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83 | ASSUME(0, optionIsSet("warn") ); |
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84 | option("nowarn"); |
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85 | ASSUME(0, 0 == optionIsSet("warn") ); |
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86 | } |
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87 | |
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88 | /////////////////////////////////////////////////////////////////////////////// |
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89 | |
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90 | proc changechar (list @L, list #) |
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91 | "USAGE: changechar(c[,r]); c=list, r=ring |
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92 | RETURN: ring R, obtained from the ring r [default: r=basering], by changing |
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93 | ringlist(r)[1] to c. |
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94 | EXAMPLE: example changechar; shows an example |
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95 | " |
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96 | { |
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97 | def save_ring=basering; |
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98 | if( size(#)==0 ) { def @r=basering; } |
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99 | if(( size(#)==1 ) and (typeof(#[1])=="ring")) { def @r=#[1]; } |
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100 | setring @r; |
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101 | list rl=ringlist(@r); |
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102 | if(defined(@L)!=voice) { def @L=fetch(save_ring,@L); } |
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103 | if (size(@L)==1) { rl[1]=@L[1];} else { rl[1]=@L;} |
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104 | def Rnew=ring(rl); |
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105 | setring save_ring; |
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106 | return(Rnew); |
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107 | } |
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108 | example |
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109 | { "EXAMPLE:"; echo = 2; |
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110 | ring rr=2,A,dp; |
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111 | ring r=0,(x,y,u,v),(dp(2),ds); |
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112 | def R=changechar(ringlist(rr)); R;""; |
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113 | def R1=changechar(32003,R); setring R1; R1; |
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114 | kill R,R1; |
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115 | } |
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116 | /////////////////////////////////////////////////////////////////////////////// |
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117 | |
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118 | proc changeord (list @o, list #) |
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119 | "USAGE: changeord(neword[,r]); newordstr=list, r=ring/qring |
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120 | RETURN: ring R, obtained from the ring r [default: r=basering], by changing |
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121 | order(r) to neword. |
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122 | If, say, neword=list(list(\"wp\",intvec(2,3)),list(list(\"dp\",1:(n-2)))); |
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123 | and if the ring r exists and has n variables, the ring R will be |
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124 | equipped with the monomial ordering wp(2,3),dp. |
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125 | EXAMPLE: example changeord; shows an example |
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126 | " |
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127 | { |
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128 | def save_ring=basering; |
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129 | if( size(#)==0 ) { def @r=basering; } |
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130 | if( size(#)==1 ) { def @r=#[1]; } |
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131 | setring @r; |
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132 | list rl=ringlist(@r); |
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133 | rl[3]=@o; |
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134 | def Rnew=ring(rl); |
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135 | setring save_ring; |
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136 | return(Rnew); |
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137 | } |
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138 | example |
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139 | { "EXAMPLE:"; echo = 2; |
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140 | ring r=0,(x,y,u,v),(dp(2),ds); |
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141 | def R=changeord(list(list("wp",intvec(2,3)),list("dp",1:2))); R; ""; |
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142 | ideal i = x^2,y^2-u^3,v; |
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143 | qring Q = std(i); |
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144 | def Q'=changeord(list(list("lp",nvars(Q))),Q); setring Q'; Q'; |
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145 | kill R,Q,Q'; |
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146 | } |
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147 | /////////////////////////////////////////////////////////////////////////////// |
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148 | |
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149 | proc changevar (string vars, list #) |
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150 | "USAGE: changevar(vars[,r]); vars=string, r=ring/qring |
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151 | RETURN: ring R, obtained from the ring r [default: r=basering], by changing |
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152 | varstr(r) according to the value of vars. |
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153 | If, say, vars = \"t()\" and the ring r exists and has n |
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154 | variables, the new basering will have name R and variables |
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155 | t(1),...,t(n). |
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156 | If vars = \"a,b,c,d\", the new ring will have the variables a,b,c,d. |
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157 | NOTE: This procedure is useful in connection with the procedure ringtensor, |
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158 | when a conflict between variable names must be avoided. |
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159 | This proc uses 'execute' or calls a procedure using 'execute'. |
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160 | EXAMPLE: example changevar; shows an example |
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161 | " |
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162 | { |
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163 | if( size(#)==0 ) { def @r=basering; } |
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164 | if( size(#)==1 ) { def @r=#[1]; } |
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165 | setring @r; |
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166 | ideal i = ideal(@r); int @q = size(i); |
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167 | if( @q!=0 ) |
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168 | { string @s = "Rnew1"; } |
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169 | else |
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170 | { string @s = "Rnew"; } |
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171 | string @newring = @s+"=("+charstr(@r)+"),("; |
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172 | if( vars[size(vars)-1]=="(" and vars[size(vars)]==")" ) |
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173 | { |
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174 | @newring = @newring+vars[1,size(vars)-2]+"(1.."+string(nvars(@r))+")"; |
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175 | } |
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176 | else { @newring = @newring+vars; } |
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177 | string ords=ordstr(@r); |
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178 | int l=size(ords); |
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179 | int l1,l2; |
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180 | while(l>0) |
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181 | { |
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182 | if (ords[l]=="(") { l1=l; break; } |
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183 | if (ords[l]==")") { l2=l; } |
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184 | l--; |
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185 | } |
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186 | string last_ord=string(ords[l1-3..l1-1]); |
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187 | if ((last_ord[1]!="w") |
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188 | && (last_ord[1]!="W") |
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189 | && (last_ord[2]!="M")) |
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190 | { |
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191 | if (l2==size(ords)) { ords=string(ords[1..l1-1]); } |
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192 | else { ords=string(ords[1..l1-1])+string(ords[l2+1..size(ords)]); } |
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193 | } |
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194 | @newring = @newring+"),("+ords+");"; |
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195 | execute("ring "+@newring); |
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196 | if( @q!=0 ) |
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197 | { |
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198 | map phi = @r,maxideal(1); |
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199 | ideal i = phi(i); |
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200 | attrib(i,"isSB",1); //*** attrib funktioniert ? |
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201 | qring Rnew=i; |
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202 | } |
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203 | return(Rnew); |
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204 | } |
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205 | example |
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206 | { "EXAMPLE:"; echo = 2; |
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207 | ring r=0,(x,y,u,v),(dp(2),ds); |
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208 | ideal i = x^2,y^2-u^3,v; |
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209 | qring Q = std(i); |
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210 | setring(r); |
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211 | def R=changevar("A()"); R; ""; |
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212 | def Q'=changevar("a,b,c,d",Q); setring Q'; Q'; |
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213 | kill R,Q,Q'; |
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214 | } |
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215 | /////////////////////////////////////////////////////////////////////////////// |
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216 | |
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217 | proc defring (string s2, int n, string s3, string s4) |
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218 | "USAGE: defring(ch,n,va,or); ch,va,or=strings, n=integer |
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219 | RETURN: ring R with characteristic 'ch', ordering 'or' and n variables with |
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220 | names derived from va. |
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221 | If va is a single letter, say va=\"a\", and if n<=26 then a and the |
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222 | following n-1 letters from the alphabet (cyclic order) are taken as |
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223 | variables. If n>26 or if va is a single letter followed by a bracket, |
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224 | say va=\"T(\", the variables are T(1),...,T(n). |
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225 | NOTE: This proc is useful for defining a ring in a procedure. |
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226 | This proc uses 'execute' or calls a procedure using 'execute'. |
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227 | EXAMPLE: example defring; shows an example |
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228 | " |
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229 | { |
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230 | string @newring = "ring newring =("+s2+"),("; |
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231 | if( n>26 or s3[2]=="(" ) { string @v = s3[1]+"(1.."+string(n)+")"; } |
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232 | else { string @v = A_Z(s3,n); } |
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233 | @newring=@newring+@v+"),("+s4+");"; |
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234 | execute(@newring); |
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235 | return(newring); |
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236 | } |
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237 | example |
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238 | { "EXAMPLE:"; echo = 2; |
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239 | def r=defring("0",5,"u","ls"); r; setring r;""; |
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240 | def R=defring("2,A",10,"x(","dp(3),ws(1,2,3),ds"); R; setring R; |
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241 | kill R,r; |
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242 | } |
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243 | /////////////////////////////////////////////////////////////////////////////// |
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244 | |
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245 | proc defrings (int n, list #) |
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246 | "USAGE: defrings(n,[p]); n,p integers |
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247 | RETURN: ring R with characteristic p [default: p=32003], ordering ds and n |
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248 | variables x,y,z,a,b,...if n<=26 (resp. x(1..n) if n>26) |
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249 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
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250 | EXAMPLE: example defrings; shows an example |
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251 | " |
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252 | { |
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253 | int p; |
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254 | if (size(#)==0) { p=32003; } |
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255 | else { p=#[1]; } |
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256 | if (n >26) |
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257 | { |
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258 | string s="ring S ="+string(p)+",x(1.."+string(n)+"),ds;"; |
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259 | } |
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260 | else |
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261 | { |
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262 | string s="ring S ="+string(p)+",("+A_Z("x",n)+"),ds;"; |
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263 | } |
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264 | execute(s); |
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265 | dbprint(printlevel-voice+2," |
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266 | // 'defrings' created a ring. To see the ring, type (if the name R was |
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267 | // assigned to the return value): |
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268 | show R; |
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269 | // To make the ring the active basering, type |
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270 | setring R; "); |
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271 | return(S); |
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272 | } |
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273 | example |
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274 | { "EXAMPLE:"; echo = 2; |
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275 | def S5=defrings(5,0); S5; ""; |
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276 | def S30=defrings(30); S30; |
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277 | kill S5,S30; |
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278 | } |
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279 | /////////////////////////////////////////////////////////////////////////////// |
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280 | |
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281 | proc defringp (int n,list #) |
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282 | "USAGE: defringp(n,[p]); n,p=integers |
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283 | RETURN: ring R with characteristic p [default: p=32003], ordering dp and n |
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284 | variables x,y,z,a,b,...if n<=26 (resp. x(1..n) if n>26) |
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285 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
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286 | EXAMPLE: example defringp; shows an example |
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287 | " |
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288 | { |
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289 | int p; |
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290 | if (size(#)==0) { p=32003; } |
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291 | else { p=#[1]; } |
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292 | if (n >26) |
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293 | { |
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294 | string s="ring P="+string(p)+",x(1.."+string(n)+"),dp;"; |
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295 | } |
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296 | else |
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297 | { |
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298 | string s="ring P="+string(p)+",("+A_Z("x",n)+"),dp;"; |
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299 | } |
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300 | execute(s); |
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301 | dbprint(printlevel-voice+2," |
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302 | // 'defringp' created a ring. To see the ring, type (if the name R was |
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303 | // assigned to the return value): |
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304 | show R; |
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305 | // To make the ring the active basering, type |
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306 | setring R; "); |
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307 | return(P); |
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308 | } |
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309 | example |
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310 | { "EXAMPLE:"; echo = 2; |
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311 | def P5=defringp(5,0); P5; ""; |
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312 | def P30=defringp(30); P30; |
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313 | kill P5,P30; |
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314 | } |
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315 | /////////////////////////////////////////////////////////////////////////////// |
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316 | |
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317 | proc extendring (int n, string va, string o, list #) |
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318 | "USAGE: extendring(n,va,o[,iv,i,r]); va,o=strings, n,i=integers, r=ring, |
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319 | iv=intvec of positive integers or iv=0 |
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320 | RETURN: ring R, which extends the ring r by adding n new variables in front |
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321 | of (resp. after, if i!=0) the old variables. |
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322 | [default: (i,r)=(0,basering)]. |
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323 | @* -- The characteristic is the characteristic of r. |
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324 | @* -- The new vars are derived from va. If va is a single letter, say |
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325 | va=\"T\", and if n<=26 then T and the following n-1 letters from |
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326 | T..Z..T (resp. T(1..n) if n>26) are taken as additional variables. |
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327 | If va is a single letter followed by a bracket, say va=\"x(\", |
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328 | the new variables are x(1),...,x(n). |
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329 | @* -- The ordering is the product ordering of the ordering of r and of an |
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330 | ordering derived from `o` [and iv]. |
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331 | @* - If o contains a 'c' or a 'C' in front resp. at the end, this is |
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332 | taken for the whole ordering in front, resp. at the end. If o does |
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333 | not contain a 'c' or a 'C' the same rule applies to ordstr(r). |
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334 | @* - If no intvec iv is given, or if iv=0, o may be any allowed ordstr, |
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335 | like \"ds\" or \"dp(2),wp(1,2,3),Ds(2)\" or \"ds(a),dp(b),ls\" if |
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336 | a and b are globally (!) defined integers and if a+b+1<=n. |
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337 | If, however, a and b are local to a proc calling extendring, the |
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338 | intvec iv must be used to let extendring know the values of a and b |
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339 | @* - If a non-zero intvec iv is given, iv[1],iv[2],... are taken for the |
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340 | 1st, 2nd,... block of o, if o contains no substring \"w\" or \"W\" |
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341 | i.e. no weighted ordering (in the above case o=\"ds,dp,ls\" |
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342 | and iv=a,b). |
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343 | If o contains a weighted ordering (only one (!) weighted block is |
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344 | allowed) iv[1] is taken as size for the weight-vector, the next |
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345 | iv[1] values of iv are taken as weights and the remaining values of |
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346 | iv as block size for the remaining non-weighted blocks. |
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347 | e.g. o=\"dp,ws,Dp,ds\", iv=3,2,3,4,2,5 creates the ordering |
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348 | dp(2),ws(2,3,4),Dp(5),ds |
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349 | NOTE: This proc is useful for adding deformation parameters. |
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350 | This proc uses 'execute' or calls a procedure using 'execute'. |
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351 | If you use it in your own proc, it may be advisable to let the local |
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352 | names of your proc start with a @ |
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353 | EXAMPLE: example extendring; shows an example |
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354 | " |
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355 | { |
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356 | //--------------- initialization and place c/C of ordering properly ----------- |
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357 | string @o1,@o2,@ro,@wstr,@v,@newring; |
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358 | int @i,@w,@ii,@k; |
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359 | intvec @iv,@iw; |
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360 | if( find(o,"c")+find(o,"C") != 0) |
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361 | { |
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362 | @k=1; |
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363 | if( o[1]=="c" or o[1]=="C" ) { @o1=o[1,2]; o=o[3..size(o)]; } |
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364 | else { @o2=o[size(o)-1,2]; o=o[1..size(o)-2]; } |
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365 | } |
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366 | if( size(#)==0 ) { #[1]=0; } |
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367 | if( typeof(#[1])!="intvec" ) |
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368 | { |
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369 | if( size(#)==1 ) { @i=#[1]; def @r=basering; } |
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370 | if( size(#)==2 ) { @i=#[1]; def @r=#[2]; } |
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371 | if( o[size(o)]!=")" and find(o,",")==0 ) { o=o+"("+string(n)+")"; } |
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372 | } |
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373 | else |
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374 | { |
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375 | @iv=#[1]; |
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376 | if( size(#)==2 ) { @i=#[2]; def @r=basering; } |
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377 | if( size(#)==3 ) { @i=#[2]; def @r=#[3]; } |
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378 | if( @iv==0 && o[size(o)]!=")" && find(o,",")==0 ) {o=o+"("+string(n)+")";} |
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379 | } |
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380 | @ro=ordstr(@r); |
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381 | if( @ro[1]=="c" or @ro[1]=="C" ) |
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382 | { @v=@ro[1,2]; @ro=@ro[3..size(@ro)]; } |
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383 | else |
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384 | { @wstr=@ro[size(@ro)-1,2]; @ro=@ro[1..size(@ro)-2]; } |
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385 | if( @k==0) { @o1=@v; @o2=@wstr; } |
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386 | //----------------- prepare ordering if an intvec is given -------------------- |
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387 | if( typeof(#[1])=="intvec" and #[1]!=0 ) |
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388 | { |
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389 | @k=n; //@k counts no of vars not yet ordered |
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390 | @w=find(o,"w")+find(o,"W");o=o+" "; |
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391 | if( @w!=0 ) |
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392 | { |
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393 | @wstr=o[@w..@w+1]; |
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394 | o=o[1,@w-1]+"@"+o[@w+2,size(o)]; |
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395 | @iw=@iv[2..@iv[1]+1]; |
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396 | @wstr=@wstr+"("+string(@iw)+")"; |
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397 | @k=@k-@iv[1]; |
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398 | @iv=@iv[@iv[1]+2..size(@iv)]; |
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399 | @w=0; |
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400 | } |
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401 | for( @ii=1; @ii<=size(@iv); @ii=@ii+1 ) |
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402 | { |
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403 | if( find(o,",",@w+1)!=0 ) |
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404 | { |
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405 | @w=find(o,",",@w+1); |
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406 | if( o[@w-1]!="@" ) |
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407 | { |
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408 | o=o[1,@w-1]+"("+string(@iv[@ii])+")"+o[@w,size(o)]; |
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409 | @w=find(o,",",@w+1); |
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410 | @k=@k-@iv[@ii]; |
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411 | } |
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412 | else { @ii=@ii-1; } |
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413 | } |
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414 | } |
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415 | @w=find(o,"@"); |
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416 | if( @w!=0 ) { o=o[1,@w-1] + @wstr + o[@w+1,size(o)]; } |
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417 | if( @k>0 and o[size(o)]!=")" ) { o=o+"("+string(@k)+")"; } |
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418 | } |
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419 | //------------------------ prepare string of new ring ------------------------- |
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420 | @newring = "ring na =("+charstr(@r)+"),("; |
---|
421 | if( n>26 or va[2]=="(" ) { @v = va[1]+"(1.."+string(n)+")"; } |
---|
422 | else { @v = A_Z(va,n); } |
---|
423 | if( @i==0 ) |
---|
424 | { |
---|
425 | @v=@v+","+varstr(@r); |
---|
426 | o=@o1+o+","+@ro+@o2; |
---|
427 | } |
---|
428 | else |
---|
429 | { |
---|
430 | @v=varstr(@r)+","+@v; |
---|
431 | o=@o1+@ro+","+o+@o2; |
---|
432 | } |
---|
433 | @newring=@newring+@v+"),("+o+");"; |
---|
434 | //---------------------------- execute and export ----------------------------- |
---|
435 | execute(@newring); |
---|
436 | dbprint(printlevel-voice+2," |
---|
437 | // 'extendring' created a new ring. |
---|
438 | // To see the ring, type (if the name 'R' was assigned to the return value): |
---|
439 | show(R); |
---|
440 | "); |
---|
441 | |
---|
442 | return(na); |
---|
443 | } |
---|
444 | example |
---|
445 | { "EXAMPLE:"; echo = 2; |
---|
446 | ring r=0,(x,y,z),ds; |
---|
447 | show(r);""; |
---|
448 | // blocksize is derived from no of vars: |
---|
449 | int t=5; |
---|
450 | def R1=extendring(t,"a","dp"); //t global: "dp" -> "dp(5)" |
---|
451 | show(R1); setring R1; ""; |
---|
452 | def R2=extendring(4,"T(","c,dp",1,r); //"dp" -> "c,..,dp(4)" |
---|
453 | show(R2); setring R2; ""; |
---|
454 | |
---|
455 | // no intvec given, blocksize given: given blocksize is used: |
---|
456 | def R3=extendring(4,"T(","dp(2)",0,r); // "dp(2)" -> "dp(2)" |
---|
457 | show(R3); setring R3; ""; |
---|
458 | |
---|
459 | // intvec given: weights and blocksize is derived from given intvec |
---|
460 | // (no specification of a blocksize in the given ordstr is allowed!) |
---|
461 | // if intvec does not cover all given blocks, the last block is used |
---|
462 | // for the remaining variables, if intvec has too many components, |
---|
463 | // the last ones are ignored |
---|
464 | intvec v=3,2,3,4,1,3; |
---|
465 | def R4=extendring(10,"A","ds,ws,Dp,dp",v,0,r); |
---|
466 | // v covers 3 blocks: v[1] (=3) : no of components of ws |
---|
467 | // next v[1] values (=v[2..4]) give weights |
---|
468 | // remaining components of v are used for the remaining blocks |
---|
469 | show(R4); |
---|
470 | kill r,R1,R2,R3,R4; |
---|
471 | } |
---|
472 | /////////////////////////////////////////////////////////////////////////////// |
---|
473 | |
---|
474 | proc fetchall (def R, list #) |
---|
475 | "USAGE: fetchall(R[,s]); R=ring/qring, s=string |
---|
476 | CREATE: fetch all objects of ring R (of type poly/ideal/vector/module/number/matrix) |
---|
477 | into the basering. |
---|
478 | If no 2nd argument is present, the names are the same as in R. If, |
---|
479 | say, f is a polynomial in R and the 2nd argument is the string \"R\", then f |
---|
480 | is mapped to f_R etc. |
---|
481 | RETURN: no return value |
---|
482 | NOTE: As fetch, this procedure maps the 1st, 2nd, ... variable of R to the |
---|
483 | 1st, 2nd, ... variable of the basering. |
---|
484 | The 2nd argument is useful in order to avoid conflicts of names, the |
---|
485 | empty string is allowed |
---|
486 | CAUTION: fetchall does not work for locally defined names. |
---|
487 | It does not work if R contains a map. |
---|
488 | SEE ALSO: imapall |
---|
489 | EXAMPLE: example fetchall; shows an example |
---|
490 | " |
---|
491 | { |
---|
492 | list @L@=names(R); |
---|
493 | int @ii@; string @s@; |
---|
494 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
495 | for( @ii@=size(@L@); @ii@>0; @ii@-- ) |
---|
496 | { |
---|
497 | execute("def "+@L@[@ii@]+@s@+"=fetch(R,`@L@[@ii@]`);"); |
---|
498 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
499 | } |
---|
500 | return(); |
---|
501 | } |
---|
502 | example |
---|
503 | { "EXAMPLE:"; echo=2; |
---|
504 | // The example is not shown since fetchall does not work in a procedure; |
---|
505 | // (and hence not in the example procedure). Try the following commands: |
---|
506 | // ring R=0,(x,y,z),dp; |
---|
507 | // ideal j=x,y2,z2; |
---|
508 | // matrix M[2][3]=1,2,3,x,y,z; |
---|
509 | // j; print(M); |
---|
510 | // ring S=0,(a,b,c),ds; |
---|
511 | // fetchall(R); //map from R to S: x->a, y->b, z->c; |
---|
512 | // names(S); |
---|
513 | // j; print(M); |
---|
514 | // fetchall(S,"1"); //identity map of S: copy objects, change names |
---|
515 | // names(S); |
---|
516 | // kill R,S; |
---|
517 | } |
---|
518 | /////////////////////////////////////////////////////////////////////////////// |
---|
519 | |
---|
520 | proc imapall (def R, list #) |
---|
521 | "USAGE: imapall(R[,s]); R=ring/qring, s=string |
---|
522 | CREATE: map all objects of ring R (of type poly/ideal/vector/module/number/matrix) |
---|
523 | into the basering by applying imap to all objects of R. |
---|
524 | If no 2nd argument is present, the names are the same as in R. If, |
---|
525 | say, f is a polynomial in R and the 3rd argument is the string \"R\", then f |
---|
526 | is mapped to f_R etc. |
---|
527 | RETURN: no return value |
---|
528 | NOTE: As imap, this procedure maps the variables of R to the variables with |
---|
529 | the same name in the basering, the other variables are mapped to 0. |
---|
530 | The 2nd argument is useful in order to avoid conflicts of names, the |
---|
531 | empty string is allowed |
---|
532 | CAUTION: imapall does not work for locally defined names. |
---|
533 | It does not work if R contains a map |
---|
534 | SEE ALSO: fetchall |
---|
535 | EXAMPLE: example imapall; shows an example |
---|
536 | " |
---|
537 | { |
---|
538 | list @L@=names(R); |
---|
539 | int @ii@; string @s@; |
---|
540 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
541 | for( @ii@=size(@L@); @ii@>0; @ii@-- ) |
---|
542 | { |
---|
543 | execute("def "+@L@[@ii@]+@s@+"=imap(R,`@L@[@ii@]`);"); |
---|
544 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
545 | } |
---|
546 | return(); |
---|
547 | } |
---|
548 | example |
---|
549 | { "EXAMPLE:"; echo = 2; |
---|
550 | // The example is not shown since imapall does not work in a procedure |
---|
551 | // (and hence not in the example procedure). Try the following commands: |
---|
552 | // ring R=0,(x,y,z,u),dp; |
---|
553 | // ideal j=x,y,z,u2+ux+z; |
---|
554 | // matrix M[2][3]=1,2,3,x,y,uz; |
---|
555 | // j; print(M); |
---|
556 | // ring S=0,(a,b,c,x,z,y),ds; |
---|
557 | // imapall(R); //map from R to S: x->x, y->y, z->z, u->0 |
---|
558 | // names(S); |
---|
559 | // j; print(M); |
---|
560 | // imapall(S,"1"); //identity map of S: copy objects, change names |
---|
561 | // names(S); |
---|
562 | // kill R,S; |
---|
563 | } |
---|
564 | /////////////////////////////////////////////////////////////////////////////// |
---|
565 | |
---|
566 | proc mapall (def R, ideal i, list #) |
---|
567 | "USAGE: mapall(R,i[,s]); R=ring/qring, i=ideal of basering, s=string |
---|
568 | CREATE: map all objects of ring R (of type poly/ideal/vector/module/number/ |
---|
569 | matrix, map) into the basering by mapping the j-th variable of R to |
---|
570 | the j-th generator of the ideal i. If no 3rd argument is present, the |
---|
571 | names are the same as in R. If, say, f is a polynomial in R and the 3rd |
---|
572 | argument is the string \"R\", then f is mapped to f_R etc. |
---|
573 | RETURN: no return value. |
---|
574 | NOTE: This procedure has the same effect as defining a map, say psi, by |
---|
575 | map psi=R,i; and then applying psi to all objects of R. In particular, |
---|
576 | maps from R to some ring S are composed with psi, creating thus a map |
---|
577 | from the basering to S. |
---|
578 | mapall may be combined with copyring to change vars for all objects. |
---|
579 | The 3rd argument is useful in order to avoid conflicts of names, the |
---|
580 | empty string is allowed. |
---|
581 | CAUTION: mapall does not work for locally defined names. |
---|
582 | EXAMPLE: example mapall; shows an example |
---|
583 | " |
---|
584 | { |
---|
585 | list @L@=names(R); map @psi@ = R,i; |
---|
586 | int @ii@; string @s@; |
---|
587 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
588 | for( @ii@=size(@L@); @ii@>0; @ii@-- ) |
---|
589 | { |
---|
590 | execute("def "+@L@[@ii@]+@s@+"=@psi@(`@L@[@ii@]`);"); |
---|
591 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
592 | } |
---|
593 | return(); |
---|
594 | } |
---|
595 | example |
---|
596 | { "EXAMPLE:"; echo = 2; |
---|
597 | // The example is not shown since mapall does not work in a procedure |
---|
598 | // (and hence not in the example procedure). Try the following commands: |
---|
599 | // ring R=0,(x,y,z),dp; |
---|
600 | // ideal j=x,y,z; |
---|
601 | // matrix M[2][3]=1,2,3,x,y,z; |
---|
602 | // map phi=R,x2,y2,z2; |
---|
603 | // ring S=0,(a,b,c),ds; |
---|
604 | // ideal i=c,a,b; |
---|
605 | // mapall(R,i); //map from R to S: x->c, y->a, z->b |
---|
606 | // names(S); |
---|
607 | // j; print(M); phi; //phi maps R to S: x->c2, y->a2, z->b2 |
---|
608 | // ideal i1=a2,a+b,1; |
---|
609 | // mapall(R,i1,\"\"); //map from R to S: x->a2, y->a+b, z->1 |
---|
610 | // names(S); |
---|
611 | // j_; print(M_); phi_; |
---|
612 | // changevar(\"T\",\"x()\",R); //change vars in R and call result T |
---|
613 | // mapall(R,maxideal(1)); //identity map from R to T |
---|
614 | // names(T); |
---|
615 | // j; print(M); phi; |
---|
616 | // kill R,S,T; |
---|
617 | } |
---|
618 | /////////////////////////////////////////////////////////////////////////////// |
---|
619 | |
---|
620 | proc ord_test (def r) |
---|
621 | "USAGE: ord_test(r); r ring/qring |
---|
622 | RETURN: int 1 (resp. -1, resp. 0) if ordering of r is global (resp. local, |
---|
623 | resp. mixed) |
---|
624 | SEE ALSO: attrib |
---|
625 | EXAMPLE: example ord_test; shows an example |
---|
626 | " |
---|
627 | { |
---|
628 | if (typeof(r) != "ring") |
---|
629 | { |
---|
630 | ERROR("ord_test requires a ring/qring as input"); |
---|
631 | } |
---|
632 | if (attrib(r,"global")==1) { return(1);} |
---|
633 | def BAS = basering; |
---|
634 | setring r; |
---|
635 | poly f; |
---|
636 | int n,o,u = nvars(r),1,1; |
---|
637 | int ii; |
---|
638 | for ( ii=1; ii<=n; ii++ ) |
---|
639 | { |
---|
640 | f = 1+var(ii); |
---|
641 | o = o*(lead(f) == var(ii)); |
---|
642 | u = u*(lead(f) == 1); |
---|
643 | } |
---|
644 | setring BAS; |
---|
645 | if ( o==1 ) { return(1); } |
---|
646 | if ( u==1 ) { return(-1); } |
---|
647 | else { return(0); } |
---|
648 | } |
---|
649 | example |
---|
650 | { "EXAMPLE:"; echo = 2; |
---|
651 | ring R = 0,(x,y),dp; |
---|
652 | ring S = 0,(u,v),ls; |
---|
653 | ord_test(R); |
---|
654 | ord_test(S); |
---|
655 | ord_test(R+S); |
---|
656 | } |
---|
657 | /////////////////////////////////////////////////////////////////////////////// |
---|
658 | |
---|
659 | proc ringtensor (list #) |
---|
660 | "USAGE: ringtensor(r1,r2,...); r1,r2,...=rings |
---|
661 | RETURN: ring R whose variables are the variables from all rings r1,r2,... |
---|
662 | and whose monomial ordering is the block (product) ordering of the |
---|
663 | respective monomial orderings of r1,r2,... . Hence, R |
---|
664 | is the tensor product of the rings r1,r2,... with ordering matrix |
---|
665 | equal to the direct sum of the ordering matrices of r1,r2,... |
---|
666 | NOTE: The characteristic of the new ring will be p if one ring has |
---|
667 | characteristic p. The names of variables in the rings r1,r2,... |
---|
668 | must differ. |
---|
669 | The procedure works also for quotient rings ri, if the characteristic |
---|
670 | of ri is compatible with the characteristic of the result |
---|
671 | (i.e. if imap from ri to the result is implemented) |
---|
672 | SEE ALSO: ring operations |
---|
673 | EXAMPLE: example ringtensor; shows an example |
---|
674 | " |
---|
675 | { |
---|
676 | int @i; |
---|
677 | int @n = size(#); |
---|
678 | if (@n<=1) { ERROR("at least 2 rings required"); } |
---|
679 | def @s=#[1]+#[2]; |
---|
680 | for (@i=3; @i<=@n;@i++) |
---|
681 | { |
---|
682 | def @ss=@s+#[@i]; |
---|
683 | kill @s; |
---|
684 | def @s=@ss; |
---|
685 | kill @ss; |
---|
686 | } |
---|
687 | dbprint(printlevel-voice+2," |
---|
688 | // 'ringtensor' created a ring. To see the ring, type (if the name R was |
---|
689 | // assigned to the return value): |
---|
690 | show(R); |
---|
691 | // To make the ring the active basering, type |
---|
692 | setring R; "); |
---|
693 | return(@s); |
---|
694 | } |
---|
695 | example |
---|
696 | { "EXAMPLE:"; echo = 2; |
---|
697 | ring r=32003,(x,y,u,v),dp; |
---|
698 | ring s=0,(a,b,c),wp(1,2,3); |
---|
699 | ring t=0,x(1..5),(c,ls); |
---|
700 | def R=ringtensor(r,s,t); |
---|
701 | type R; |
---|
702 | setring s; |
---|
703 | ideal i = a2+b3+c5; |
---|
704 | def S=changevar("x,y,z"); //change vars of s |
---|
705 | setring S; |
---|
706 | qring qS =std(fetch(s,i)); //create qring of S mod i (mapped to S) |
---|
707 | def T=changevar("d,e,f,g,h",t); //change vars of t |
---|
708 | setring T; |
---|
709 | qring qT=std(d2+e2-f3); //create qring of T mod d2+e2-f3 |
---|
710 | def Q=ringtensor(s,qS,t,qT); |
---|
711 | setring Q; type Q; |
---|
712 | kill R,S,T,Q; |
---|
713 | } |
---|
714 | /////////////////////////////////////////////////////////////////////////////// |
---|
715 | |
---|
716 | proc ringweights (def P) |
---|
717 | "USAGE: ringweights(P); P=name of an existing ring (true name, not a string) |
---|
718 | RETURN: intvec consisting of the weights of the variables of P, as they |
---|
719 | appear when typing P;. |
---|
720 | NOTE: This is useful when enlarging P but keeping the weights of the old |
---|
721 | variables. |
---|
722 | EXAMPLE: example ringweights; shows an example |
---|
723 | " |
---|
724 | { |
---|
725 | int i; |
---|
726 | intvec rw; |
---|
727 | //------------------------- find weights ------------------------- |
---|
728 | for(i=nvars(P);i>0;i--) |
---|
729 | { rw[i]=ord(var(i)); } |
---|
730 | return(rw); |
---|
731 | } |
---|
732 | example |
---|
733 | {"EXAMPLE:"; echo = 2; |
---|
734 | ring r0 = 0,(x,y,z),dp; |
---|
735 | ringweights(r0); |
---|
736 | ring r1 = 0,x(1..5),(ds(3),wp(2,3)); |
---|
737 | ringweights(r1);""; |
---|
738 | // an example for enlarging the ring, keeping the first weights: |
---|
739 | intvec v = ringweights(r1),6,2,3,4,5; |
---|
740 | ring R = 0,x(1..10),(a(v),dp); |
---|
741 | ordstr(R); |
---|
742 | } |
---|
743 | /////////////////////////////////////////////////////////////////////////////// |
---|
744 | proc preimageLoc(string R_name,string phi_name,string Q_name ) |
---|
745 | "USAGE: preimageLoc ( ring_name, map_name, ideal_name ); |
---|
746 | all input parameters of type string |
---|
747 | RETURN: ideal |
---|
748 | PURPOSE: compute the preimage of an ideal under a given map for non-global |
---|
749 | orderings. |
---|
750 | The 2nd argument has to be the name of a map from the basering to |
---|
751 | the given ring (or the name of an ideal defining such a map), and |
---|
752 | the ideal has to be an ideal in the given ring. |
---|
753 | SEE ALSO: preimage |
---|
754 | KEYWORDS: preimage under a map between local rings, map between local rings, map between local and global rings |
---|
755 | EXAMPLE: example preimageLoc ; shows an example |
---|
756 | "{ |
---|
757 | def S=basering; |
---|
758 | int i; |
---|
759 | string newRing,minpoly_string; |
---|
760 | if(attrib(S,"global")!=1) |
---|
761 | { |
---|
762 | if(size(ideal(S))>0) /*qring*/ |
---|
763 | { |
---|
764 | ideal I=ideal(S); |
---|
765 | newRing="ring S0=("+charstr(S)+"),("+varstr(S)+"),dp;"; |
---|
766 | minpoly_string=string(minpoly); |
---|
767 | execute(newRing); |
---|
768 | execute("minpoly="+minpoly_string+";"); |
---|
769 | ideal I=imap(S,I); |
---|
770 | list pr=primdecGTZ(I); |
---|
771 | newRing="ring SL=("+charstr(S)+"),("+varstr(S)+"),("+ordstr(S)+");"; |
---|
772 | execute(newRing); |
---|
773 | execute("minpoly="+minpoly_string+";"); |
---|
774 | list pr=imap(S0,pr); |
---|
775 | ideal I0=std(pr[1][1]); |
---|
776 | for(i=2;i<=size(pr);i++) |
---|
777 | { |
---|
778 | I0=intersect(I0,std(pr[i][1])); |
---|
779 | } |
---|
780 | setring S0; |
---|
781 | ideal I0=imap(SL,I0); |
---|
782 | qring S1=std(I0); |
---|
783 | } |
---|
784 | else |
---|
785 | { |
---|
786 | def S1=S; |
---|
787 | } |
---|
788 | } |
---|
789 | else |
---|
790 | { |
---|
791 | def S1=S; |
---|
792 | } |
---|
793 | def @R=`R_name`; |
---|
794 | setring @R; |
---|
795 | def @phi=`phi_name`; |
---|
796 | ideal phiId=ideal(@phi); |
---|
797 | def Q=`Q_name`; |
---|
798 | if(attrib(@R,"global")!=1) |
---|
799 | { |
---|
800 | if(size(ideal(@R))>0) /*qring*/ |
---|
801 | { |
---|
802 | ideal J=ideal(@R); |
---|
803 | newRing="ring R0=("+charstr(@R)+"),("+varstr(@R)+"),dp;"; |
---|
804 | minpoly_string=string(minpoly); |
---|
805 | execute(newRing); |
---|
806 | execute("minpoly="+minpoly_string+";"); |
---|
807 | ideal J=imap(@R,J); |
---|
808 | list pr=primdecGTZ(J); |
---|
809 | newRing="ring RL=("+charstr(@R)+"),("+varstr(@R)+"),("+ordstr(@R)+");"; |
---|
810 | execute(newRing); |
---|
811 | execute("minpoly="+minpoly_string+";"); |
---|
812 | list pr=imap(R0,pr); |
---|
813 | ideal J0=std(pr[1][1]); |
---|
814 | for(i=2;i<=size(pr);i++) |
---|
815 | { |
---|
816 | J0=intersect(J0,std(pr[i][1])); |
---|
817 | } |
---|
818 | setring R0; |
---|
819 | ideal J0=imap(RL,J0); |
---|
820 | qring R1=std(J0); |
---|
821 | ideal Q=imap(@R,Q); |
---|
822 | map @phi=S1,imap(@R,phiId); |
---|
823 | } |
---|
824 | else |
---|
825 | { |
---|
826 | def R1=@R; |
---|
827 | } |
---|
828 | } |
---|
829 | else |
---|
830 | { |
---|
831 | def R1=@R; |
---|
832 | } |
---|
833 | setring S1; |
---|
834 | ideal preQ=preimage(R1,@phi,Q); |
---|
835 | setring S; |
---|
836 | ideal prQ=imap(S1,preQ); |
---|
837 | return(prQ); |
---|
838 | } |
---|
839 | example |
---|
840 | { "EXAMPLE:"; echo=2; |
---|
841 | ring S =0,(x,y,z),dp; |
---|
842 | ring R0=0,(x,y,z),ds; |
---|
843 | qring R=std(x+x2); |
---|
844 | map psi=S,x,y,z; |
---|
845 | ideal null; |
---|
846 | setring S; |
---|
847 | ideal nu=preimageLoc("R","psi","null"); |
---|
848 | nu; |
---|
849 | } |
---|
850 | |
---|
851 | ////////////////////////////////////////////////////////////////////////////// |
---|
852 | /* moved here from the nctools.lib */ |
---|
853 | /////////////////////////////////////////////////////////////////////////////// |
---|
854 | proc rootofUnity(int n) |
---|
855 | "USAGE: rootofUnity(n); n an integer |
---|
856 | RETURN: number |
---|
857 | PURPOSE: compute the minimal polynomial for the n-th primitive root of unity |
---|
858 | NOTE: works only in field extensions by one element |
---|
859 | EXAMPLE: example rootofUnity; shows examples |
---|
860 | " |
---|
861 | { |
---|
862 | if ( npars(basering) !=1 ) |
---|
863 | { |
---|
864 | ERROR(" the procedure works only with one ring parameter variable"); |
---|
865 | } |
---|
866 | if (n<0) { ERROR(" cannot compute ("+string(n)+")-th primitive root of unity"); } |
---|
867 | if (n==0) { return(number(0));} |
---|
868 | number mp = par(1); |
---|
869 | if (n==1) { return(mp-1); } |
---|
870 | if (n==2) { return(mp+1); } |
---|
871 | def OldRing = basering; |
---|
872 | string CH = charstr(basering); |
---|
873 | string MCH; |
---|
874 | int j=1; |
---|
875 | while ( (CH[j] !=",") && (j<=size(CH))) |
---|
876 | { |
---|
877 | MCH=MCH+CH[j]; j++; |
---|
878 | } |
---|
879 | string SR = "ring @@rR="+MCH+","+parstr(basering)+",dp;"; |
---|
880 | execute(SR); |
---|
881 | poly @t=var(1)^n-1; // (x^2i-1)=(x^i-1)(x^i+1) |
---|
882 | list l=factorize(@t); |
---|
883 | ideal @l=l[1]; |
---|
884 | list @d; |
---|
885 | int s=size(@l); |
---|
886 | int d=deg(@l[s]); |
---|
887 | int cnt=1; |
---|
888 | poly res; |
---|
889 | for (j=s-1; j>=1; j--) |
---|
890 | { |
---|
891 | if ( deg(@l[j]) > d) { d=deg(@l[j]); } |
---|
892 | } |
---|
893 | for (j=1; j<=s; j++) |
---|
894 | { |
---|
895 | if ( deg(@l[j]) == d) { @d[cnt]=@l[j]; cnt++; } |
---|
896 | } |
---|
897 | |
---|
898 | j=1; |
---|
899 | int i; |
---|
900 | number pw; |
---|
901 | |
---|
902 | int @sized = size(@d); |
---|
903 | |
---|
904 | if (@sized==1) |
---|
905 | { |
---|
906 | setring OldRing; |
---|
907 | list @rl = imap(@@rR,@d); |
---|
908 | mp = number(@rl[1]); |
---|
909 | kill @@rR; |
---|
910 | return(mp); |
---|
911 | } |
---|
912 | def @rng; |
---|
913 | |
---|
914 | setring OldRing; |
---|
915 | |
---|
916 | list rl = ringlist( OldRing); |
---|
917 | while ( j<=@sized ) |
---|
918 | { |
---|
919 | ASSUME(0, n%2 ==0); |
---|
920 | setring OldRing; |
---|
921 | @rng = ring(rl); |
---|
922 | setring @rng; |
---|
923 | list @rl = imap(@@rR,@d); |
---|
924 | number mp = leadcoef( @rl[j] ); |
---|
925 | minpoly = mp; |
---|
926 | number mp = minpoly; |
---|
927 | number pw = par(1)^(n div 2); |
---|
928 | if ( (pw != 1) || n==1 ) { break; } |
---|
929 | j = j+1; |
---|
930 | } |
---|
931 | setring OldRing; |
---|
932 | list @rl=imap(@@rR,@d); |
---|
933 | mp = leadcoef( @rl[j] ); |
---|
934 | kill @@rR; |
---|
935 | return(mp); |
---|
936 | } |
---|
937 | example |
---|
938 | { |
---|
939 | "EXAMPLE:";echo=2; |
---|
940 | ring r = (0,q),(x,y,z),dp; |
---|
941 | rootofUnity(6); |
---|
942 | rootofUnity(7); |
---|
943 | minpoly = rootofUnity(8); |
---|
944 | r; |
---|
945 | } |
---|
946 | |
---|
947 | proc isQuotientRing(def rng ) |
---|
948 | "USAGE: isQuotientRing ( rng ); |
---|
949 | RETURN: 1 if rng is a quotient ring, 0 otherwise. |
---|
950 | PURPOSE: check if typeof a rng "qring" |
---|
951 | KEYWORDS: qring ring ideal 'factor ring' |
---|
952 | EXAMPLE: example isQuotientRing ; shows an example |
---|
953 | " |
---|
954 | { |
---|
955 | if ( defined(basering) ) { def BAS=basering; } |
---|
956 | else { return (0); } |
---|
957 | |
---|
958 | //access to quotient ideal will fail, if basering and rng differs. |
---|
959 | setring rng; |
---|
960 | int result = ( size(ideal(rng)) != 0); |
---|
961 | |
---|
962 | setring BAS; |
---|
963 | return (result); |
---|
964 | } |
---|
965 | example |
---|
966 | { |
---|
967 | "EXAMPLE:";echo=2; |
---|
968 | ring rng = 0,x,dp; |
---|
969 | isQuotientRing(rng); //no |
---|
970 | // if a certain method does not support quotient rings, |
---|
971 | // then a parameter test could be performed: |
---|
972 | ASSUME( 0, 0==isQuotientRing(basering)); |
---|
973 | |
---|
974 | qring q= ideal(x); // constructs rng/ideal(x) |
---|
975 | isQuotientRing(q); // yes |
---|
976 | } |
---|
977 | |
---|
978 | static proc testIsQuotientRing() |
---|
979 | { |
---|
980 | ring rng7 = 7, x, dp; |
---|
981 | |
---|
982 | ring rng = real,x,dp; |
---|
983 | ASSUME(0, 0== isQuotientRing(rng) ) ; |
---|
984 | ASSUME(0, 0== isQuotientRing(rng7) ) ; |
---|
985 | ASSUME(0, char(basering)==0); // check that basering was not changed |
---|
986 | |
---|
987 | qring qrng = 1; |
---|
988 | ASSUME(0, isQuotientRing(qrng) ) ; |
---|
989 | |
---|
990 | ring rng2 = integer,x,dp; |
---|
991 | ASSUME(0, 0 == isQuotientRing(rng2) ) ; |
---|
992 | |
---|
993 | qring qrng2=0; |
---|
994 | ASSUME(0, not isQuotientRing(qrng2) ) ; |
---|
995 | |
---|
996 | ring rng3 = 0,x,dp; |
---|
997 | ASSUME(0, 0 == isQuotientRing(rng3) ) ; |
---|
998 | |
---|
999 | qring qrng3=1; |
---|
1000 | ASSUME(0, isQuotientRing(qrng3) ) ; |
---|
1001 | } |
---|
1002 | |
---|
1003 | proc hasFieldCoefficient(def rng ) |
---|
1004 | "USAGE: hasFieldCoefficient ( rng ); |
---|
1005 | RETURN: 1 if the coefficients form (and are considered to be) a field, 0 otherwise. |
---|
1006 | KEYWORDS: ring coefficients |
---|
1007 | EXAMPLE: example hasFieldCoefficient; shows an example |
---|
1008 | SEE ALSO: attrib |
---|
1009 | " |
---|
1010 | { |
---|
1011 | return (attrib(rng,"ring_cf")==0); |
---|
1012 | } |
---|
1013 | example |
---|
1014 | { |
---|
1015 | "EXAMPLE:";echo=2; |
---|
1016 | ring rng = integer,x,dp; |
---|
1017 | hasFieldCoefficient(rng); //no |
---|
1018 | // if a certain method supports only rings with integer coefficients, |
---|
1019 | // then a parameter test could be performed: |
---|
1020 | ring rng2 = 0, x, dp; |
---|
1021 | hasFieldCoefficient(rng2); // yes |
---|
1022 | } |
---|
1023 | |
---|
1024 | proc hasAlgExtensionCoefficient(def rng ) |
---|
1025 | "USAGE: hasAlgExtensionCoefficient ( rng ); |
---|
1026 | RETURN: 1 if the coeffcients are an algebraic extension, 0 otherwise. |
---|
1027 | KEYWORDS: ring coefficients |
---|
1028 | EXAMPLE: example hasAlgExtensionCoefficient; shows an example |
---|
1029 | " |
---|
1030 | { |
---|
1031 | return(attrib(rng,"cf_class")==7); |
---|
1032 | } |
---|
1033 | example |
---|
1034 | { |
---|
1035 | "EXAMPLE:";echo=2; |
---|
1036 | ring rng = integer,x,dp; |
---|
1037 | hasAlgExtensionCoefficient(rng); //no |
---|
1038 | ring rng2 = (0,a), x, dp; minpoly=a2-1; |
---|
1039 | hasAlgExtensionCoefficient(rng2); // yes |
---|
1040 | ring rng3=(49,a),x,dp; |
---|
1041 | hasAlgExtensionCoefficient(rng3); // no |
---|
1042 | } |
---|
1043 | |
---|
1044 | proc hasTransExtensionCoefficient(def rng ) |
---|
1045 | "USAGE: hasTransExtensionCoefficient ( rng ); |
---|
1046 | RETURN: 1 if the coeffcients are rational functions, 0 otherwise. |
---|
1047 | KEYWORDS: ring coefficients |
---|
1048 | EXAMPLE: example hasTransExtensionCoefficient; shows an example |
---|
1049 | " |
---|
1050 | { |
---|
1051 | return(attrib(rng,"cf_class")==8); |
---|
1052 | } |
---|
1053 | example |
---|
1054 | { |
---|
1055 | "EXAMPLE:";echo=2; |
---|
1056 | ring rng = integer,x,dp; |
---|
1057 | hasTransExtensionCoefficient(rng); //no |
---|
1058 | ring rng2 = (0,a), x, dp; |
---|
1059 | hasTransExtensionCoefficient(rng2); // yes |
---|
1060 | ring rng3=(49,a),x,dp; |
---|
1061 | hasTransExtensionCoefficient(rng3); // no |
---|
1062 | } |
---|
1063 | |
---|
1064 | proc hasGFCoefficient(def rng ) |
---|
1065 | "USAGE: hasGFCoefficient ( rng ); |
---|
1066 | RETURN: 1 if the coeffcients are of the form GF(p,k), 0 otherwise. |
---|
1067 | KEYWORDS: ring coefficients |
---|
1068 | EXAMPLE: example hasGFCoefficient; shows an example |
---|
1069 | " |
---|
1070 | { |
---|
1071 | //return((charstr(rng)!=string(char(rng))) && |
---|
1072 | //(npars(rng)==1) && |
---|
1073 | //(find(charstr(rng),string(char(rng)))!=1) && |
---|
1074 | //(charstr(basering)<>"real")&& |
---|
1075 | //(charstr(basering)<>"complex") ); |
---|
1076 | return(attrib(rng,"cf_class")==4); |
---|
1077 | } |
---|
1078 | example |
---|
1079 | { |
---|
1080 | "EXAMPLE:";echo=2; |
---|
1081 | ring r1 = integer,x,dp; |
---|
1082 | hasGFCoefficient(r1); |
---|
1083 | ring r2 = (4,a),x,dp; |
---|
1084 | hasGFCoefficient(r2); |
---|
1085 | ring r3 = (2,a),x,dp; |
---|
1086 | minpoly=a2+a+1; |
---|
1087 | hasGFCoefficient(r2); |
---|
1088 | } |
---|
1089 | |
---|
1090 | proc hasZp_aCoefficient(def rng ) |
---|
1091 | "USAGE: hasZp_aCoefficient ( rng ); |
---|
1092 | RETURN: 1 if the coeffcients are of the form Zp_a(p,k), 0 otherwise. |
---|
1093 | KEYWORDS: ring coefficients |
---|
1094 | EXAMPLE: example hasZp_aCoefficient; shows an example |
---|
1095 | " |
---|
1096 | { |
---|
1097 | return((attrib(rng,"cf_class")==7) and (char(rng)>0)); |
---|
1098 | } |
---|
1099 | example |
---|
1100 | { |
---|
1101 | "EXAMPLE:";echo=2; |
---|
1102 | ring r1 = integer,x,dp; |
---|
1103 | hasZp_aCoefficient(r1); |
---|
1104 | ring r2 = (4,a),x,dp; |
---|
1105 | hasZp_aCoefficient(r2); |
---|
1106 | ring r3 = (2,a),x,dp; |
---|
1107 | minpoly=a2+a+1; |
---|
1108 | hasZp_aCoefficient(r2); |
---|
1109 | } |
---|
1110 | |
---|
1111 | proc hasZpCoefficient(def rng ) |
---|
1112 | "USAGE: hasZpCoefficient ( rng ); |
---|
1113 | RETURN: 1 if the coeffcients are of the form ZZ/p, 0 otherwise. |
---|
1114 | KEYWORDS: ring coefficients |
---|
1115 | EXAMPLE: example hasZpCoefficient; shows an example |
---|
1116 | " |
---|
1117 | { |
---|
1118 | return(attrib(rng,"cf_class")==1); |
---|
1119 | } |
---|
1120 | example |
---|
1121 | { |
---|
1122 | "EXAMPLE:";echo=2; |
---|
1123 | ring r1 = integer,x,dp; |
---|
1124 | hasZpCoefficient(r1); |
---|
1125 | ring r2 = 7,x,dp; |
---|
1126 | hasZpCoefficient(r2); |
---|
1127 | } |
---|
1128 | |
---|
1129 | proc hasQQCoefficient(def rng ) |
---|
1130 | "USAGE: hasQQCoefficient ( rng ); |
---|
1131 | RETURN: 1 if the coeffcients are QQ, 0 otherwise. |
---|
1132 | KEYWORDS: ring coefficients |
---|
1133 | EXAMPLE: example hasQQCoefficient; shows an example |
---|
1134 | " |
---|
1135 | { |
---|
1136 | return(attrib(rng,"cf_class")==2); |
---|
1137 | } |
---|
1138 | example |
---|
1139 | { |
---|
1140 | "EXAMPLE:";echo=2; |
---|
1141 | ring r1 = integer,x,dp; |
---|
1142 | hasQQCoefficient(r1); |
---|
1143 | ring r2 = QQ,x,dp; |
---|
1144 | hasQQCoefficient(r2); |
---|
1145 | } |
---|
1146 | |
---|
1147 | proc hasGlobalOrdering (def rng) |
---|
1148 | "USAGE: hasGlobalOrdering ( rng ); |
---|
1149 | RETURN: 1 if rng has a global monomial ordering, 0 otherwise. |
---|
1150 | KEYWORDS: monomial ordering |
---|
1151 | EXAMPLE: example hasGlobalOrdering; shows an example |
---|
1152 | " |
---|
1153 | { |
---|
1154 | return (attrib(rng,"global")==1); |
---|
1155 | } |
---|
1156 | example |
---|
1157 | { |
---|
1158 | ring rng = integer,x,dp; |
---|
1159 | hasGlobalOrdering(rng); //yes |
---|
1160 | ring rng2 = 0, x, ds; |
---|
1161 | hasGlobalOrdering(rng2); // no |
---|
1162 | } |
---|
1163 | |
---|
1164 | proc hasCommutativeVars (def rng) |
---|
1165 | "USAGE: hasCommutativeVars ( rng ); |
---|
1166 | RETURN: 1 if rng is a commutative polynomial ring, 0 otherwise. |
---|
1167 | KEYWORDS: plural |
---|
1168 | EXAMPLE: example hasCommutativeVars; shows an example |
---|
1169 | " |
---|
1170 | { |
---|
1171 | list rl=ringlist(rng); |
---|
1172 | return (size(rl)==4); |
---|
1173 | } |
---|
1174 | example |
---|
1175 | { |
---|
1176 | ring r=0,(x,y,z),dp; |
---|
1177 | hasCommutativeVars(r); |
---|
1178 | } |
---|
1179 | |
---|
1180 | proc hasNumericCoeffs(def rng) |
---|
1181 | "USAGE: hasNumericCoeffs ( rng ); |
---|
1182 | RETURN: 1 if rng has inexact coeffcients, 0 otherwise. |
---|
1183 | KEYWORDS: floating point |
---|
1184 | EXAMPLE: example hasNumericCoeffs; shows an example |
---|
1185 | " |
---|
1186 | { |
---|
1187 | return((attrib(rng,"cf_class")==3) /*real*/ |
---|
1188 | or (attrib(rng,"cf_class")==5) /*gmp real*/ |
---|
1189 | or (attrib(rng,"cf_class")==9) /*gmp complex*/); |
---|
1190 | } |
---|
1191 | example |
---|
1192 | { |
---|
1193 | "EXAMPLE:";echo=2; |
---|
1194 | ring r1 = integer,x,dp; |
---|
1195 | hasNumericCoeffs(r1); |
---|
1196 | ring r2 = complex,x,dp; |
---|
1197 | hasNumericCoeffs(r2); |
---|
1198 | } |
---|
1199 | |
---|
1200 | |
---|
1201 | proc isSubModule(def I,def J) |
---|
1202 | "USAGE: isSubModule(I,J): I, J: ideal or module |
---|
1203 | RETURN: 1 if module(I) is in module(J), 0 otherwise |
---|
1204 | EXAMPLE: isSubModule; shows an example |
---|
1205 | { |
---|
1206 | if (attrib(J,"isSB")) |
---|
1207 | { return(size(reduce(I,J,5))==0); } |
---|
1208 | else |
---|
1209 | { return(size(reduce(I,groebner(J),5))==0); } |
---|
1210 | } |
---|
1211 | example |
---|
1212 | { |
---|
1213 | "EXAMPLE:"; echo = 2; |
---|
1214 | ring r=0,x,dp; |
---|
1215 | ideal I1=x2; |
---|
1216 | ideal I2=x3; |
---|
1217 | isSubModule(I1, I2); |
---|
1218 | isSubModule(I2, I1); |
---|
1219 | } |
---|
1220 | |
---|
1221 | proc hasMixedOrdering() |
---|
1222 | "USAGE: hasMixedOrdering(); |
---|
1223 | RETURN: 1 if ordering of basering is mixed, 0 else |
---|
1224 | EXAMPLE: example hasMixedOrdering(); shows an example |
---|
1225 | " |
---|
1226 | { |
---|
1227 | int i,p,m; |
---|
1228 | for(i = 1; i <= nvars(basering); i++) |
---|
1229 | { |
---|
1230 | if(var(i) > 1) |
---|
1231 | { |
---|
1232 | p++; |
---|
1233 | } |
---|
1234 | else |
---|
1235 | { |
---|
1236 | m++; |
---|
1237 | } |
---|
1238 | } |
---|
1239 | if((p > 0) && (m > 0)) { return(1); } |
---|
1240 | return(0); |
---|
1241 | } |
---|
1242 | example |
---|
1243 | { "EXAMPLE:"; echo = 2; |
---|
1244 | ring R1 = 0,(x,y,z),dp; |
---|
1245 | hasMixedOrdering(); |
---|
1246 | ring R2 = 31,(x(1..4),y(1..3)),(ds(4),lp(3)); |
---|
1247 | hasMixedOrdering(); |
---|
1248 | ring R3 = 181,x(1..9),(dp(5),lp(4)); |
---|
1249 | hasMixedOrdering(); |
---|
1250 | } |
---|
1251 | |
---|
1252 | proc changeordTo(def r,string o) |
---|
1253 | "USAGE: changeordTo(ring, string s); |
---|
1254 | RETURN: a ring with the oderinging changed to the (simple) ordering s |
---|
1255 | EXAMPLE: example changeordTo(); shows an example |
---|
1256 | " |
---|
1257 | { |
---|
1258 | list rl=ringlist(r); |
---|
1259 | rl[3]=list(list("C",0),list(o,1:nvars(r))); |
---|
1260 | def rr=ring(rl); |
---|
1261 | return(rr); |
---|
1262 | } |
---|
1263 | example |
---|
1264 | { |
---|
1265 | "EXAMPLE:"; echo = 2; |
---|
1266 | ring r=0,(x,y),lp; |
---|
1267 | def rr=changeordTo(r,"dp"); |
---|
1268 | rr; |
---|
1269 | } |
---|
1270 | |
---|
1271 | proc addvarsTo(def r,list vars,int blockorder) |
---|
1272 | "USAGE: addvarsTo(ring,list_of_strings, int); |
---|
1273 | int may be: 0:ordering: dp |
---|
1274 | 1:ordering dp,dp |
---|
1275 | 2:oring.ordering,dp |
---|
1276 | RETURN: a ring with the addtional variables |
---|
1277 | EXAMPLE: example addvarsTo(); shows an example |
---|
1278 | " |
---|
1279 | { |
---|
1280 | list rl=ringlist(r); |
---|
1281 | int n=nvars(r); |
---|
1282 | rl[2]=rl[2]+vars; |
---|
1283 | if (blockorder==0) |
---|
1284 | { |
---|
1285 | rl[3]=list(list("C",0),list("dp",1:(nvars(r)+size(vars)))); |
---|
1286 | } |
---|
1287 | else |
---|
1288 | { |
---|
1289 | if (blockorder==2) |
---|
1290 | { |
---|
1291 | rl[3]=rl[3]+list(list("dp",1:size(vars))); |
---|
1292 | } |
---|
1293 | else |
---|
1294 | { |
---|
1295 | rl[3]=list(list("C",0),list("dp",1:nvars(r)),list("dp",1:size(vars))); |
---|
1296 | } |
---|
1297 | } |
---|
1298 | def rr=ring(rl); |
---|
1299 | return(rr); |
---|
1300 | } |
---|
1301 | example |
---|
1302 | { |
---|
1303 | "EXAMPLE:"; echo = 2; |
---|
1304 | ring r=0,(x,y),lp; |
---|
1305 | def rr=addvarsTo(r,list("a","b"),0); |
---|
1306 | rr; kill rr; |
---|
1307 | def rr=addvarsTo(r,list("a","b"),1); |
---|
1308 | rr; kill rr; |
---|
1309 | def rr=addvarsTo(r,list("a","b"),2); |
---|
1310 | rr; |
---|
1311 | } |
---|
1312 | proc addNvarsTo(def r,int N,string n,int blockorder) |
---|
1313 | "USAGE: addNvarsTo(ring,int N, string name, int b); |
---|
1314 | b may be: 0:ordering: dp |
---|
1315 | 1:ordering dp,dp |
---|
1316 | 2:oring.ordering,dp |
---|
1317 | RETURN: a ring with N addtional variables |
---|
1318 | EXAMPLE: example addNvarsTo(); shows an example |
---|
1319 | " |
---|
1320 | { |
---|
1321 | list v; |
---|
1322 | for(int i=N;i>0;i--) { v[i]=n+"("+string(i)+")"; } |
---|
1323 | return(addvarsTo(r,v,blockorder)); |
---|
1324 | } |
---|
1325 | example |
---|
1326 | { |
---|
1327 | "EXAMPLE:"; echo = 2; |
---|
1328 | ring r=0,(x,y),lp; |
---|
1329 | def rr=addNvarsTo(r,2,"@",0); |
---|
1330 | rr; kill rr; |
---|
1331 | def rr=addNvarsTo(r,2,"@",1); |
---|
1332 | rr; kill rr; |
---|
1333 | def rr=addNvarsTo(r,2,"@",2); |
---|
1334 | rr; |
---|
1335 | } |
---|