version 1.4, 2004/09/15 07:41:59 |
version 1.6, 2004/09/30 07:39:42 |
|
|
% $OpenXM: OpenXM/src/kan96xx/Doc/gfan.sm1,v 1.3 2004/09/14 08:30:47 takayama Exp $ |
% $OpenXM: OpenXM/src/kan96xx/Doc/gfan.sm1,v 1.5 2004/09/16 06:16:44 takayama Exp $ |
% cp cone.sm1 $OpenXM_HOME/src/kan96xx/Doc/gfan.sm1 |
% cp cone.sm1 $OpenXM_HOME/src/kan96xx/Doc/gfan.sm1 |
% $Id$ |
% $Id$ |
% iso-2022-jp |
% iso-2022-jp |
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%% Two examples are given below to get a global Grobner fan and |
|
%% a local Grobner fan ; cone.sample and cone.sample2 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%% Global Grobner Fan |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%% How to input data? An example. (cf. test13.sm1) |
|
%% Modify the following or copy the /cone.sample { ... } def |
|
%% to your own file, |
|
%% edit it, and execute if by " cone.sample ; " |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
/cone.sample { |
|
cone.load.cohom |
|
% write a comment about the problem. "nl" means new line. |
|
/cone.comment [ |
|
(Toric ideal for 1-simplex x 2-simplex, in k[x]) nl |
|
] cat def |
|
|
|
% List of variables |
|
% If cone.type=1, then (H) should be added. |
|
/cone.vlist [(x11) (x12) (x13) (x21) (x22) (x23) |
|
(Dx11) (Dx12) (Dx13) (Dx21) (Dx22) (Dx23) (h)] def |
|
|
|
% List of variables in the form for define_ring. |
|
/cone.vv (x11,x12,x13,x21,x22,x23) def |
|
|
|
% If cone.type=0, then x,Dx, |
|
% If cone.type=1, then x,Dx,h,H (Doubly homogenized) |
|
% If cone.type=2, then x,Dx,h |
|
/cone.type 2 def |
|
|
|
% Set how to parametrize the weight space. |
|
% In the example below, 6 means the number of variables x11,x12,x13,x21,x22,x33 |
|
% p q parametrizeSmallFan (p >= q) : Enumerate Grobner cones in the Small |
|
% Grobner fan. |
|
% The weights for the last p-q variables |
|
% are 0. |
|
% Example. 6 2 parametrizeSmallFan weights for x12,x21,x22,x23 are 0. |
|
% |
|
% p q parametrizeTotalFan (p = q = number of variables in cone.vv) |
|
% p > q has not yet been implemented. |
|
% |
|
/cone.parametrizeWeightSpace { |
|
6 6 parametrizeSmallFan |
|
} def |
|
|
|
% If you want to enumerate Grobner cones in local order (i.e., x^e <= 0), |
|
% then cone.local = 1 else cone.local = 0. |
|
/cone.local 0 def |
|
|
|
% Initial value of the weight in the weight space of which dimension is |
|
% cone.m |
|
% If it is null, then a random weight is used. |
|
/cone.w_start |
|
null |
|
def |
|
|
|
% If cone.h0=1, then the weight for h is 0. |
|
% It is usally set to 1. |
|
/cone.h0 1 def |
|
|
|
% Set input polynomials which generate the ideal. |
|
% Input must be homogenized. |
|
% (see also data/test14.sm1 for double homogenization.) |
|
/cone.input |
|
[ |
|
(x11 x22 - x12 x21) |
|
(x12 x23 - x13 x22) |
|
(x11 x23 - x13 x21) |
|
] |
|
def |
|
|
|
% Set a function to compute Grobner basis. |
|
% cone.gb_Dh : For computing in Homogenized Weyl algebra h[1,1](D). |
|
% cone.gb_DhH : For computing in doubly homogenized Weyl algebra. |
|
% ( Computation in ^O and h[0,1](^D) need this |
|
% as the first step. /cone.local 1 def ) |
|
/cone.gb { |
|
cone.gb_Dh |
|
} def |
|
|
|
|
|
cone.comment message |
|
(cone.input = ) message |
|
cone.input message |
|
%%%% Step 1. Enumerating the Grobner Cones in a global ring. |
|
%%%% The result is stored in cone.fan |
|
getGrobnerFan |
|
|
|
%%%% If you want to print the output, then uncomment. |
|
printGrobnerFan |
|
|
|
%%%% If you want to save the data to the file sm1out.txt, then uncomment. |
|
% /cone.wightGblist 1 def saveGrobnerFan /ff set ff output |
|
|
|
%%%% Step 2. Dehomogenize the Grobner Cones |
|
%%%% by the equivalence relation in a local ring (uncomment). |
|
% dhCones_h |
|
|
|
%%%% Generate the final data dhcone2.fan (a list of local Grobner cones.) |
|
% dhcone.rtable |
|
|
|
%%%% Output dhcone2.fan with explanations |
|
% dhcone.printGrobnerFan |
|
|
|
} def |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%% End of " How to input data? An example. " |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%% Local Grobner Fan |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%% How to input data? The example 2 (cf. test14.sm1). |
|
%% Modify the following or copy the /cone.sample2 { ... } def |
|
%% to your own file, |
|
%% edit it, and execute if by " cone.sample2 ; " |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
/cone.sample2 { |
|
cone.load.cohom |
|
% write a comment about the problem. "nl" means new line. |
|
/cone.comment [ |
|
(BS for y and y-(x-1)^2, t1, t2 space, in doubly homogenized Weyl algebra.) nl |
|
(The Grobner cones are dehomogenized to get local Grobner fan.) nl |
|
] cat def |
|
|
|
% List of variables |
|
% If cone.type=1, then (H) should be added. |
|
/cone.vlist [(t1) (t2) (x) (y) (Dt1) (Dt2) (Dx) (Dy) (h) (H)] def |
|
|
|
% List of variables in the form for define_ring. |
|
/cone.vv (t1,t2,x,y) def |
|
|
|
% If cone.type=0, then x,Dx, |
|
% If cone.type=1, then x,Dx,h,H (Doubly homogenized) |
|
% If cone.type=2, then x,Dx,h |
|
/cone.type 1 def |
|
|
|
% Set how to parametrize the weight space. |
|
% In the example below, 6 means the number of variables x11,x12,x13,x21,x22,x33 |
|
% p q parametrizeSmallFan (p >= q) : Enumerate Grobner cones in the Small |
|
% Grobner fan. |
|
% The weights for the last p-q variables |
|
% are 0. |
|
% Example. 6 2 parametrizeSmallFan weights for x12,x21,x22,x23 are 0. |
|
% |
|
% p q parametrizeTotalFan (p = q = number of variables in cone.vv) |
|
% p > q has not yet been implemented. |
|
% |
|
/cone.parametrizeWeightSpace { |
|
4 2 parametrizeSmallFan |
|
} def |
|
|
|
% If you want to enumerate Grobner cones in local order (i.e., x^e <= 0), |
|
% then cone.local = 1 else cone.local = 0. |
|
/cone.local 1 def |
|
|
|
% Initial value of the weight in the weight space of which dimension is |
|
% cone.m |
|
% If it is null, then a random weight is used. |
|
/cone.w_start |
|
null |
|
def |
|
|
|
% If cone.h0=1, then the weight for h is 0. |
|
% It is usally set to 1. |
|
/cone.h0 1 def |
|
|
|
% Set input polynomials which generate the ideal. |
|
% Input must be homogenized. |
|
% (see also data/test14.sm1 for double homogenization.) |
|
/cone.input |
|
[ |
|
(t1-y) (t2 - (y-(x-1)^2)) |
|
((-2 x + 2)*Dt2+Dx) |
|
(Dt1+Dt2+Dy) |
|
] |
|
def |
|
% homogenize |
|
[cone.vv ring_of_differential_operators |
|
[[(t1) -1 (t2) -1 (Dt1) 1 (Dt2) 1]] ecart.weight_vector |
|
0] define_ring |
|
dh.begin |
|
cone.input { . homogenize toString } map /cone.input set |
|
dh.end |
|
|
|
% Set a function to compute Grobner basis. |
|
% cone.gb_Dh : For computing in Homogenized Weyl algebra h[1,1](D). |
|
% cone.gb_DhH : For computing in doubly homogenized Weyl algebra. |
|
% ( Computation in ^O and h[0,1](^D) need this |
|
% as the first step. /cone.local 1 def ) |
|
/cone.gb { |
|
cone.gb_DhH |
|
} def |
|
|
|
cone.comment message |
|
(cone.input = ) message |
|
cone.input message |
|
%%%% Step 1. Enumerating the Grobner Cones in a global ring. |
|
%%%% The result is stored in cone.fan |
|
getGrobnerFan |
|
|
|
%%%% If you want to print the output, then uncomment. |
|
printGrobnerFan |
|
|
|
%%%% If you want to save the data to the file sm1out.txt, then uncomment. |
|
% /cone.wightGblist 1 def saveGrobnerFan /ff set ff output |
|
|
|
%%%% Step 2. Dehomogenize the Grobner Cones |
|
%%%% by the equivalence relation in a local ring (uncomment). |
|
dhCones_h |
|
|
|
%%%% Generate the final data dhcone2.fan (a list of local Grobner cones.) |
|
dhcone.rtable |
|
|
|
%%%% Output dhcone2.fan with explanations |
|
dhcone.printGrobnerFan |
|
|
|
} def |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%% End of " How to input data? The example 2. " |
|
%% Do not touch below. |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
|
|
[(parse) (cgi.sm1) pushfile] extension |
|
|
|
% If you use local polymake, then comment out. |
|
% If you use the cgi/polymake on the net, then uncomment out. |
|
%/doPolymake {doPolymake.OoHG} def |
|
|
|
% My setting. |
|
[(getenv) (HOST)] extension /cone.hostname set |
|
cone.hostname tag 0 eq { /cone.hostname (?) def } { } ifelse |
|
cone.hostname (orange2.math.sci.kobe-u.ac.jp) eq |
|
cone.hostname (orange2-clone.math.sci.kobe-u.ac.jp) eq |
|
or |
|
{ |
|
(Using doPolymake.OoHG ) message |
|
/doPolymake {doPolymake.OoHG} def |
|
} { } ifelse |
|
|
/cone.debug 1 def |
/cone.debug 1 def |
|
|
/ox.k0.loaded boundp { |
/ox.k0.loaded boundp { |
|
|
[(parse) (ox.sm1) pushfile] extension |
[(parse) (ox.sm1) pushfile] extension |
} ifelse |
} ifelse |
|
|
|
/cone.load.cohom { |
|
/cone.loaded boundp { } |
|
{ |
|
[(parse) (cohom.sm1) pushfile] extension |
|
[(parse) (cone.sm1) pushfile] extension |
|
[(parse) (dhecart.sm1) pushfile] extension |
|
/cone.loaded 1 def |
|
oxNoX polymake.start ( ) message |
|
} ifelse |
|
} def |
|
|
|
%% Usages: cone.gb_DhH. h H (double homogenized) $BMQ$N(B GB. |
|
%% dhecart.sm1 $B$r(B load $B$7$F$"$k$3$H(B. $BF~NO$OF1<!$G$J$$$H$$$1$J$$(B. |
|
%% [cone.vv ring_of_differential_operators |
|
%% [[(t1) -1 (t2) -1 (Dt1) 1 (Dt2) 1]] ecart.weight_vector |
|
%% 0] define_ring |
|
%% dh.begin homogenize dh.end $B$J$I$NJ}K!$GF1<!2=$G$-$k(B. |
|
/cone.gb_DhH { |
|
/arg2 set /arg1 set |
|
[/ff /ww] pushVariables |
|
[ |
|
/ff arg1 def |
|
/ww arg2 def |
|
/dh.gb.verbose 1 def |
|
/dh.autoHomogenize 0 def |
|
[(AutoReduce) 1] system_variable |
|
[ff { toString } map cone.vv |
|
[ww cone.vv generateD1_1]] dh.gb 0 get /arg1 set |
|
] pop |
|
arg1 |
|
} def |
|
|
% |
% |
% cone.fan, cone.gblist $B$K(B fan $B$N%G!<%?$,$O$$$k(B. |
% cone.fan, cone.gblist $B$K(B fan $B$N%G!<%?$,$O$$$k(B. |
% |
% |
|
%%%%<<<< $B=i4|%G!<%?$N@_DjNc(B. $BF|K\8lHG(B data/test13 $B$h$j(B. <<<<<<<<<<<<<< |
%%%%<<<< $B=i4|%G!<%?$N@_DjNc(B data/test13 $B$h$j(B. <<<<<<<<<<<<<< |
/cone.sample.test13.ja { |
/cone.sample.test13 { |
|
/cone.loaded boundp { } |
/cone.loaded boundp { } |
{ |
{ |
[(parse) (cohom.sm1) pushfile] extension |
[(parse) (cohom.sm1) pushfile] extension |
|
|
[(cones) [ ] cones] arrayToTree |
[(cones) [ ] cones] arrayToTree |
[(facets) [ ] facets] arrayToTree |
[(facets) [ ] facets] arrayToTree |
[(merged) [ ] merged] arrayToTree |
[(merged) [ ] merged] arrayToTree |
[(nextcid) [ ] merged] arrayToTree |
[(nextcid) [ ] nextcid] arrayToTree |
[(nextfid) [ ] merged] arrayToTree |
[(nextfid) [ ] nextfid] arrayToTree |
[(coneid) [ ] merged] arrayToTree |
[(coneid) [ ] coneid] arrayToTree |
] |
] |
] arrayToTree /arg1 set |
] arrayToTree /arg1 set |
] pop |
] pop |
|
|
dhcone.fan |
dhcone.fan |
} def |
} def |
|
|
% Todo: print, save functions. Representative of weight & init. |
%< |
|
% Usages: dhcone.rtable |
|
% dhcone $B$NHV9f$H(B cone $B$NHV9f$N(B $BCV49I=$r@8@.$7(B dhcone2.fan (merge $B$7$?(B cone $B$N>pJs(B) |
|
% $B$r(B dhcone.fan $B$+$i:n$k(B. dhcone2.gblist $B$b:n$kJd=u4X?t(B. |
|
% dhCones_h $B$7$F$+$i(B dhcone.rable $B$9$k(B. |
|
%> |
|
/dhcone.rtable { |
|
[/i /j /vv /cones /facets /facets2 /merged /nextcid /nextcid2 /ii /ww] pushVariables |
|
[ |
|
% $BCV49I=(B dhcone.h2dh $B$r:n$k(B. |
|
/dhcone.h2dh cone.fan length newVector.with-1 def |
|
0 1 , dhcone.fan length 1 sub { |
|
/i set |
|
dhcone.fan i get , (cones) getNode 2 get /vv set |
|
0 1 vv length 1 sub { |
|
/j set |
|
dhcone.h2dh , vv j get , i to_univNum , put |
|
} for |
|
} for |
|
% merge $B$7$?(B dhcone $B$r@0M}$7$?$b$N(B, dhcone2.fan $B$r:n$k(B. |
|
/dhcone2.fan dhcone.fan length newVector def |
|
0 1 , dhcone.fan length 1 sub { |
|
/i set |
|
dhcone.fan i get (facets) getNode 2 get /facets set |
|
dhcone.fan i get (merged) getNode 2 get /merged set |
|
dhcone.fan i get (nextcid) getNode 2 get /nextcid set |
|
dhcone.fan i get (cones) getNode 2 get /cones set |
|
/facets2 [ ] def |
|
/nextcid2 [ ] def |
|
0 1 , facets length 1 sub { |
|
/j set |
|
merged j get , (3).. eq { |
|
facets2 [ facets j get ] join /facets2 set |
|
% $B$H$J$j$N(B cone $B$,$"$k$H$-(B $BJQ49I=$K$7$?$,$$(B, cone $BHV9f$rJQ49(B |
|
nextcid2 [ dhcone.h2dh , nextcid j get , get ] join /nextcid2 set |
|
} { } ifelse |
|
merged j get , (2).. eq { |
|
facets2 [ facets j get ] join /facets2 set |
|
% $B6-3&$N$H$-(B -2 $B$rF~$l$k(B. |
|
nextcid2 [ (-2).. ] join /nextcid2 set |
|
} { } ifelse |
|
} for |
|
|
|
dhcone2.fan i , |
|
[(dhcone) [ ] |
|
[ |
|
[(facets) [ ] facets2] arrayToTree |
|
[(nextcid) [ ] nextcid2] arrayToTree |
|
[(cones) [ ] cones] arrayToTree |
|
] |
|
] arrayToTree , put |
|
|
|
} for |
|
|
|
% $B:G8e$K(B dhcone2.gblist $B$r:n$k(B. |
|
/dhcone2.gblist , dhcone2.fan length newVector , def |
|
0 1 , dhcone2.fan length 1 sub { |
|
/i set |
|
dhcone2.fan i get (cones) getNode 2 get /cones set |
|
cone.grGblist , cones 0 get , get , /ii set % GB of initial (H->1). |
|
cone.gblist i get , (weight) getNode , [ 2 0 2 ] get /ww set |
|
|
|
dhcone2.gblist i, |
|
[(gbasis) [ ] |
|
[ |
|
[(initial) [ ] ii] arrayToTree |
|
[(weight) [ ] ww] arrayToTree |
|
] |
|
] arrayToTree , put |
|
|
|
} for |
|
(dhcone2.fan, dhcone2.gblist, dhcone.h2dh are set.) message |
|
|
|
] pop |
|
popVariables |
|
} def |
|
|
|
%< |
|
% $BI=$N8+J}$N2r@b$r0u:~$9$k4X?t(B. |
|
% Usages: dhcone.explain |
|
%> |
|
/dhcone.explain { |
|
[ |
|
( ) nl |
|
(Data format in << dhcone2.fan >>, which is a dehomogenized Grobner fan.) nl nl |
|
(<< cone.vlist >> is the list of the variables.) nl |
|
@@@.Hsymbol ( is the homogenization variable to be dehomogenized.) nl nl |
|
(<< cone.input >> is generators of a given ideal.) nl nl |
|
(<< cone.d >> is the dimension of parametrization space of the weights P_w) nl |
|
( P_w is a cone in R^m where the number m is stored in << cone.m >>) nl |
|
( P_w --- W ---> R^n [weight space]. ) nl |
|
( W is stored in << cone.W >> ) nl |
|
( << u cone.W mul >> gives the weight vector standing for u) nl nl |
|
(All cones in the data lie in the weight parametrization space P_w.) nl |
|
( "facets" are the inner normal vector of the cone. ) nl |
|
( "nextcid" is a list of the cone id's of the adjacent cones.) nl |
|
( -2 in "nextcid" means that this facet lies on the border of the weight space.) nl |
|
( "cones" is a list of the cone id's of the NON-dehomonized Grobner fan) nl |
|
( stored in << cone.fan >>) nl |
|
] cat |
|
} def |
|
|
|
%< |
|
% dhcone.printGrobnerFan |
|
% dhcone $B$N0u:~4X?t(B |
|
%> |
|
/dhcone.printGrobnerFan { |
|
[/i] pushVariables |
|
[ |
|
(========== Grobner Fan (for dehomogenized cones) ============) message |
|
[ |
|
(cone.comment) |
|
(cone.vlist) (cone.vv) |
|
(cone.input) |
|
(cone.type) (cone.local) (cone.h0) |
|
(cone.n) (cone.m) (cone.d) |
|
(cone.W) (cone.Wpos) (cone.Wt) |
|
(cone.L) (cone.Lp) (cone.Lpt) |
|
(cone.weightBorder) |
|
(cone.incidence) |
|
] { printGrobnerFan.1 } map |
|
( ) message |
|
(The number of cones = ) messagen dhcone.fan length message |
|
( ) message |
|
0 1 dhcone2.fan length 1 sub { |
|
/ii set |
|
ii messagen ( : ) messagen |
|
dhcone2.fan ii get printTree |
|
} for |
|
1 { |
|
0 1 dhcone2.gblist length 1 sub { |
|
/ii set |
|
ii messagen ( : ) messagen |
|
dhcone2.gblist ii get printTree |
|
} for |
|
} { } ifelse |
|
|
|
|
|
(=========================================) message |
|
%(cone.withGblist = ) messagen cone.withGblist message |
|
dhcone.explain message |
|
( ) message |
|
] pop |
|
popVariables |
|
} def |
|
|
|
% |
|
% $B;n$7J}(B test14, 22, 25 |
|
% |
|
% (data/test14.sm1) run (data/test14-data.sm1) run |
|
% printGrobnerFan ; % H $BIU$-$G0u:~(B. |
|
% dhCones_h ; % dehomogenize Cones. |
|
% dhcone.rtable ; % dhcone2.fan $BEy$r@8@.(B. |
|
% dhcone.printGrobnerFan ; % $B0u:~(B. |
|
% $B0u:~$7$?$b$N$O(B test*-print.txt $B$X3JG<$7$F$"$k(B. |
|
% |
|
|
|
% Todo: save functions. |