Annotation of OpenXM/src/k097/lib/minimal/debug-note.txt, Revision 1.2
1.2 ! takayama 1: $OpenXM: OpenXM/src/k097/lib/minimal/debug-note.txt,v 1.1 2000/05/06 07:58:37 takayama Exp $
! 2:
! 3: minimal.k $B$O(B V-minimal free resolution $B$r9=@.$9$k(B
! 4: $B%W%m%0%i%`$G(B openxm version 1.1.2 $B0J>e$GF0:n(B.
! 5: ( $BI,MW$J(B component $B$O(B k0, ox_asir )
! 6: openxm $B$K$D$$$F$O(B, http://www.openxm.org $B$r;2>H(B.
! 7:
! 8: $B8=:_(B, $B$$$A$*$&(B error $B$J$/$H$^$j(B, V-minimal free resolution
! 9: $B$i$7$-$b$N$r9=@.$9$k$H$$$&$@$1$G(B, $B?t3XE*$J@5$7$5$N%A%'%C%/$O(B
! 10: $B$^$@(B.
! 11:
! 12: $B;H$$J}(B
! 13:
! 14: k0 ( k0 $B%$%s%?%W%j%?$r%9%?!<%H(B )
! 15: load["minimal.k"];; (minimal.k $B$r%m!<%I(B)
! 16:
! 17: $BNc(B 1: Sminimal_v $B$O(B, V-minimal free resolution $B$r(B, Schreyer resolution
! 18: $B$rJQ7A$7$F$$$C$F5a$a$k(B. (Sminimal $B$O(B LaScala-Stillman's algorithm
! 19: $B$r;H$&(B: $B$^$@(B negative weight vector $B$G$-$A$s$H$&$4$+$J$$(B.)
! 20:
! 21: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
! 22: v=[[2*x*Dx + 3*y*Dy+6, 0],
! 23: [3*x^2*Dy + 2*y*Dx, 0],
! 24: [0, x^2+y^2],
! 25: [0, x*y]];
! 26: a=Sminimal_v(v);
! 27: sm1_pmat(a[0]); b=a[0]; b[1]*b[0]:
! 28:
! 29: $B%N!<%H(B: a[0] is the V-minimal resolution. a[3] is the Schreyer resolution.
! 30:
! 31: $BNc(B 2:
! 32: a=Sannfs3("x^3-y^2*z^2");
! 33: b=a[0]; sm1_pmat(b);
! 34: b[1]*b[0]: b[2]*b[1]: ===> complex $B$G$"$k$3$H$N$?$7$+$a(B.
! 35:
! 36: x^3-y^2*z^2 $B$N(B annihilating ideal $B$N(B laplace $BJQ49$N(B V-minimal free resolution.
! 37: Weight $B$O(B (-1,-1,-1,1,1,1).
! 38:
! 39: $B$A$J$_$K(B,
! 40: Map(a[3],"Length"): $B$O(B 8, 17, 13, 3 (Schreyer resolution $B$N(B betti $B?t(B).
! 41: Map(a[0],"Length"): $B$O(B 4, 6, 2 (V-minimal resolution $B$N(B betti $B?t(B).
! 42:
! 43:
! 44:
! 45: ------- $B%F%9%H%G!<%?=8(B
! 46:
1.1 takayama 47: a=Sannfs2("x*y*(x-y)*(x+y)");
48:
1.2 ! takayama 49: a=testAnnfs3("x*y*z*(x+y+z-1)");
! 50: V-minimal $B$K$b(B 1 $B$,@.J,$H$7$F$N$3$k$b$N$"$j(B.
! 51:
! 52: a=testAnnfs2("x^3-y^2-x-1");
! 53:
! 54: a=testAnnfs3("x^3+y^3+z^3");
! 55: Schreyer $B$N(B betti $B$O(B max 100 $BDxEY(B.
! 56: incompatible ... $B$J$k(B error $B$,$G$k$1$I$$$$$+!)(B
! 57: Warning in order.c: mmLarger_tower3(): incompatible input and gbList.
! 58:
! 59: Length of gb is 6, f is es, g is -es^6*Dy^2
! 60: Warning in order.c: mmLarger_tower3(): incompatible input and gbList.
! 61: 20 $BJ,8e(B segmentation fault $B$G=*N;(B.
! 62:
! 63:
1.1 takayama 64:
65:
66: -------- successful construction x^3-y^2-x
67: def Sannfs2_laScala(f) {
68: local p,pp;
69: p = Sannfs(f,"x,y");
70: /* Do not make laplace transform. */
71: sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");
72:
73: #define TOTAL_STRATEGY
74:
75: % k0
76: sm1>macro package : dr.sm1, 9/26,1995 --- Version 2/2, 2000.
77: sm1>macro package : module1.sm1, 1994 -- Nov 8, 1998
78: This is kan/k0 Version 1998,12/15
79: WARNING: This is an EXPERIMENTAL version
80: sm1>var.sm1 : Version 3/7, 1997
81:
82:
83: In(1)=Loading startup files (startup.k) 1997, 3/11.
84: sm1 version = 3.000320
85: Default ring is Z[x,h].
86: WARNING(sm): You rewrited the protected symbol pushVariables.
87: WARNING(sm): You rewrited the protected symbol popVariables.
88: In(2)=a=Sannfs2_laScala("x^3-y^2-x");
89:
90: %Warning: The identifier <<Sannfs2_laScala>> is not in the system dictionary
91: % nor in the user dictionaries. Push NullObject.
92: ERROR(sm): Warning: identifier is not in the dictionaries
93: --- Engine error or interrupt : The error occured on the top level.
94: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
95: In(3)=load["minimal.k"];;
96: cpp: -lang-c++: linker input file unused since linking not done
97: --- Engine error or interrupt : The error occured on the top level.
98: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
99: --- Engine error or interrupt : The error occured on the top level.
100: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
101: cohom.sm1 is the top of an experimental package to compute restrictions
102: of all degrees based on restall.sm1 and restall_s.sm1
103: See, http://www.math.kobe-u.ac.jp to get these files of the latest version.
104: Note that the package b-function.sm1 cannot be used with this package.
105: r-interface.sm1 (C) N.Takayama, restriction, deRham
106:
107: oxasir.sm1, --- open asir protocol module 3/1 1998, 6/5 1999
108: asirconnect, asir, fctr, primadec, (C) M.Noro, N.Takayama
109: ox.sm1, --- open sm1 protocol module 11/11,1999 (C) N.Takayama. oxhelp for help
110: hol.sm1, basic package for holonomic systems (C) N.Takayama, 1999, 12/07
111: rank characteristic ch rrank gb pgb syz genericAnn annfs
112: sm1>gkz.sm1 generates gkz systems (C) N.Takayama, 1998, 11/8, cf. rrank in hol.sm1
113: gkz
114: sm1>appell.sm1 generates Appell hypergeometric differential equations (C) N.Takayama, 1998, 11/8, cf. rank in hol.sm1
115: appell1 appell4
116: sm1>resol0.sm1, package to construct schreyer resolutions -- not minimal
117: (C) N.Takayama, 1999, 5/18. resol0, resol1
118: complex.sm1 : 1999, 9/28, res-div, res-solv, res-kernel-image, res-dual
119: In this package, complex is expressed in terms of matrices.
120: restall.sm1 ... compute all the cohomology groups of the restriction
121: of a D-module to tt = (t_1,...,t_d) = (0,...,0).
122: non-Schreyer Version: 19980415 by T.Oaku
123: usage: [(P1)...] [(t1)...] bfm --> the b-function
124: [(P1)...] [(t1)...] k0 k1 deg restall --> cohomologies of restriction
125: [(P1)...] [(t1)...] intbfm --> the b-function for integration
126: [(P1)...] [(t1)...] k0 k1 deg intall --> cohomologies of integration
127: restall_s.sm1...compute all the cohomology groups of the restriction
128: of a D-module to tt = (t_1,...,t_d) = (0,...,0).
129: Schreyer Version: 19990521 by N.Takayama & T.Oaku
130: usage: [(P1)...] [(t1)...] k0 k1 deg restall_s -> cohomologies of restriction
131: [(P1)...] [(t1)...] k0 k1 deg intall_s --> cohomologies of integration
132: No truncation from below in restall
133: The variable Schreyer is set to 2.
134: Loading tower.sm1 in the standard context. You cannot use Schyrer 1. It is controlled from cohom.sm1
135:
136: /e_ $e_$. def /x $x$. def /y $y$. def /H $H$. def /E $E$. def /Dx $Dx$. def /Dy $Dy$. def /h $h$. def
137: /e_ $e_$. def /es $es$. def /x $x$. def /y $y$. def /z $z$. def /H $H$. def /E $E$. def /ES $ES$. def /Dx $Dx$. def /Dy $Dy$. def /Dz $Dz$. def /h $h$. def
138: In(4)=a=Sannfs2_laScala("x^3-y^2-x");
139: Starting ox_asir server.
140: Hello from open. serverName is localhost and portnumber is 0
141: Done the initialization. port =1146
142: Hello from open. serverName is localhost and portnumber is 0
143: Done the initialization. port =1147
144: [ 7 , 1147 , 6 , 1146 ]
145: [1] 6699
146: Trying to accept from localhost... len= 16
147: 4 7c 7f 0 0 1 0 0 0 0 0 0 0 0 8 0
148: Authentification: localhost is allowed to be accepted.
149: Accepted.
150: Trying to accept from localhost... len= 16
151: 4 7d 7f 0 0 1 0 0 0 0 0 0 0 0 6 0
152: Authentification: localhost is allowed to be accepted.
153: Accepted.
154:
155: Control port 1146 : Connected.
156:
157: Stream port 1147 : Connected.
158: Byte order for control process is network byte order.
159: Byte order for engine process is network byte order.
160: /e_ $e_$. def /es $es$. def /x $x$. def /y $y$. def /H $H$. def /E $E$. def /ES $ES$. def /Dx $Dx$. def /Dy $Dy$. def /h $h$. def
161: WeightOfSweyl=[ x , -1 , y , -1 , Dx , 1 , Dy , 1 ]
162: [ 3*y*Dx^2 , -2*x*Dx*Dy , -6*x*Dx^3 , 9*y^2*Dx*Dy^2 , 27*y^3*Dy^3 ]
163: Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
164: .......Done. betti=7
165: Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
166: Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
167: ....Done. betti=4
168: Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
169: Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
170: .Done. betti=1
171: Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
172: Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
173: Done. betti=0
174: Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
175: [ 0 , 0 ]
176: Processing [ 0 , 0 ] Strategy = 2
177: [ 0 , 1 ]
178: Processing [ 0 , 1 ] Strategy = 2
179: [ 0 , 2 ]
180: Processing [ 0 , 2 ] Strategy = 3
181: [ 1 , 2 ]
182: Processing [ 1 , 2 ] Strategy = 3
183: SpairAndReduction:
184: [ p and bases , [ [ 0 , 1 ] , [ -2*x*Dy , -3*y*Dx ] ] , [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , %[null] , %[null] ] ]
185: [ -2*x*Dy , -3*es*y*Dx ]
186: [gi, gj] = [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h ]
187: 1
188: Reduce the element 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2-6*y^2*Dx^2*h+4*x^2*Dy^2*h+2*x*y*Dy*h^2+2*x*h^4
189: by [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , %[null] , %[null] ]
190: result is [ 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , 1 , [ 2*y*h , 0 , 0 , 0 , 0 ] ]
191: vdegree of the original = 1
192: vdegree of the remainder = 1
193: [ 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , [ -2*x*Dy+2*y*h , -3*y*Dx , 0 , 0 , 0 ] , 2 , 3 , 1 , 1 ]
194: [ 1 , 3 ]
195: Processing [ 1 , 3 ] Strategy = 3
196: SpairAndReduction:
197: [ p and bases , [ [ 0 , 2 ] , [ -2*x*Dx , -y ] ] , [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , %[null] ] ]
198: [ -2*x*Dx , -es^2*y ]
199: [gi, gj] = [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 ]
200: 1
201: Reduce the element 4*x^2*Dx*Dy*h+6*x*y*Dy^2*h+4*x*Dy*h^3-4*x*y*Dx*h^2
202: by [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , %[null] ]
203: result is [ 0 , -1 , [ 0 , -2*x*h , 0 , 0 , 0 ] ]
204: vdegree of the original = 1
205: vdegree of the remainder = %[null]
206: [ 0 , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , 3 , -1 , 1 , %[null] ]
207: [ 1 , 6 ]
208: Processing [ 1 , 6 ] Strategy = 3
209: SpairAndReduction:
210: [ p and bases , [ [ 1 , 2 ] , [ -3*Dx^2 , Dy ] ] , [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , %[null] ] ]
211: [ -3*es*Dx^2 , es^2*Dy ]
212: [gi, gj] = [ -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 ]
213: 1
214: Reduce the element 9*y*Dx^2*Dy^2+18*Dx^2*Dy*h^2-6*y*Dx^3*h-6*x*Dy^3*h+6*x*Dx*Dy*h^2
215: by [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , %[null] ]
216: result is [ 0 , -1 , [ 3*Dy^2-2*Dx*h , -h^2 , 0 , 0 , 0 ] ]
217: vdegree of the original = 3
218: vdegree of the remainder = %[null]
219: [ 0 , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] , 6 , -1 , 3 , %[null] ]
220: [ 2 , 1 ]
221: Processing [ 2 , 1 ] Strategy = 3
222: SpairAndReduction:
223: [ p and bases , [ [ 2 , 3 ] , [ -Dx , Dy ] ] , [ %[null] , %[null] , [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ] , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , %[null] , %[null] , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] ] ]
224: [ es^2*Dx , -es^3*Dy ]
225: [gi, gj] = [ 2*x*Dy+3*es*y*Dx+es^3-2*y*h , 2*x*Dx+es^2*y-2*es*x*h ]
226: 1
227: Reduce the element 3*es*y*Dx^2-es^2*y*Dy+es^3*Dx-es^2*h^2+2*Dy*h^2-2*y*Dx*h+2*es*x*Dy*h
228: by [ %[null] , %[null] , [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ] , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , %[null] , %[null] , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] ]
229: result is [ -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 , 1 , [ 0 , 0 , 0 , 0 , 0 , 0 , -y ] ]
230: vdegree of the original = 2
231: vdegree of the remainder = 2
232: [ -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 , [ 0 , 0 , Dx , -Dy , 0 , 0 , -y ] , 1 , 5 , 2 , 2 ]
233: [ 0 , 3 ]
234: Processing [ 0 , 3 ] Strategy = 4
235: [ 1 , 0 ]
236: Processing [ 1 , 0 ] Strategy = 4
237: SpairAndReduction:
238: [ p and bases , [ [ 1 , 3 ] , [ 9*y^2*Dy , 2*x ] ] , [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , %[null] ] ]
239: [ 9*es*y^2*Dy , 2*es^3*x ]
240: [gi, gj] = [ -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 ]
241: 1
242: Reduce the element -27*y^3*Dy^3-12*x^2*Dx^2*h^2+24*x*y*Dx*Dy*h^2-45*y^2*Dy^2*h^2+18*y^3*Dx*Dy*h+8*x^3*Dy^2*h+18*y^2*Dx*h^3-4*x^2*y*Dy*h^2+4*x^2*h^4-4*x*y^2*h^3
243: by [ 3*y*Dx^2-2*x*Dy*h-y*h^2 , -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h , -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 , 9*y^2*Dx*Dy^2-6*x*Dx^2*h^2+12*y*Dx*Dy*h^2+4*x^2*Dy^2*h-2*x*y*Dy*h^2+2*x*h^4-2*y^2*h^3 , %[null] ]
244: result is [ 27*y^3*Dy^3+12*x^2*Dx^2*h^2+81*y^2*Dy^2*h^2+24*y*Dy*h^4-18*y^3*Dx*Dy*h-8*x^3*Dy^2*h-42*y^2*Dx*h^3+4*x^2*y*Dy*h^2-4*x^2*h^4+4*x*y^2*h^3 , -1 , [ 0 , -12*y*h^2 , 0 , 0 , 0 ] ]
245: vdegree of the original = 0
246: vdegree of the remainder = 0
247: [ 27*y^3*Dy^3+12*x^2*Dx^2*h^2+81*y^2*Dy^2*h^2+24*y*Dy*h^4-18*y^3*Dx*Dy*h-8*x^3*Dy^2*h-42*y^2*Dx*h^3+4*x^2*y*Dy*h^2-4*x^2*h^4+4*x*y^2*h^3 , [ 0 , -9*y^2*Dy-12*y*h^2 , 0 , -2*x , 0 ] , 0 , 4 , 0 , 0 ]
248: [ 1 , 5 ]
249: Processing [ 1 , 5 ] Strategy = 4
250: [ 2 , 0 ]
251: Processing [ 2 , 0 ] Strategy = 4
252: SpairAndReduction:
253: [ p and bases , [ [ 2 , 5 ] , [ 3*y*Dy , 2*x ] ] , [ [ 0 , 9*y^2*Dy+12*y*h^2 , 0 , 2*x , 1 ] , %[null] , [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ] , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , %[null] , -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] ] ]
254: [ -3*es^2*y*Dy , -2*es^5*x ]
255: [gi, gj] = [ 2*x*Dy+3*es*y*Dx+es^3-2*y*h , -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 ]
256: 1
257: Reduce the element -9*es*y^2*Dx*Dy-2*es^3*x*Dx-3*es^3*y*Dy+2*es^2*x*h^2-4*x*Dy*h^2-9*es*y*Dx*h^2+6*y^2*Dy*h-4*es*x^2*Dy*h+6*y*h^3-2*es*x*y*h^2
258: by [ [ 0 , 9*y^2*Dy+12*y*h^2 , 0 , 2*x , 1 ] , %[null] , [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ] , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , %[null] , -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] ]
259: result is [ -3*es^3*y*Dy+es^4*Dx+2*es^2*x*h^2+9*es*y*Dx*h^2+4*es^3*h^2+6*y^2*Dy*h-4*es*x^2*Dy*h+2*y*h^3-2*es*x*y*h^2 , 1 , [ Dx , 0 , 2*h^2 , 0 , 0 , 0 , 0 ] ]
260: vdegree of the original = 1
261: vdegree of the remainder = 1
262: [ -3*es^3*y*Dy+es^4*Dx+2*es^2*x*h^2+9*es*y*Dx*h^2+4*es^3*h^2+6*y^2*Dy*h-4*es*x^2*Dy*h+2*y*h^3-2*es*x*y*h^2 , [ Dx , 0 , -3*y*Dy+2*h^2 , 0 , 0 , -2*x , 0 ] , 0 , 1 , 1 , 1 ]
263: [ 3 , 0 ]
264: Processing [ 3 , 0 ] Strategy = 4
265: SpairAndReduction:
266: [ p and bases , [ [ 0 , 1 ] , [ -Dx , -3*y*Dy ] ] , [ [ -Dx , 1 , 3*y*Dy-2*h^2 , 0 , 0 , 2*x , 0 ] , [ 0 , 0 , -Dx , Dy , 0 , 1 , y ] , %[null] , %[null] ] ]
267: [ -Dx , -3*es*y*Dy ]
268: [gi, gj] = [ 3*es^2*y*Dy+2*es^5*x-Dx+es-2*es^2*h^2 , -es^2*Dx+es^3*Dy+es^6*y+es^5 ]
269: 1
270: Reduce the element -3*es^3*y*Dy^2-2*es^5*x*Dx+Dx^2-3*es^6*y^2*Dy-3*es^5*y*Dy-es*Dx+2*es^2*Dx*h^2-3*es^6*y*h^2-2*es^5*h^2
271: by [ [ -Dx , 1 , 3*y*Dy-2*h^2 , 0 , 0 , 2*x , 0 ] , [ 0 , 0 , -Dx , Dy , 0 , 1 , y ] , %[null] , %[null] ]
272: result is [ 3*es^3*y*Dy^2+2*es^5*x*Dx-Dx^2+3*es^6*y^2*Dy+3*es^5*y*Dy+es*Dx-2*es^3*Dy*h^2+es^6*y*h^2 , -1 , [ 0 , -2*h^2 , 0 , 0 ] ]
273: vdegree of the original = 2
274: vdegree of the remainder = 2
275: [ 3*es^3*y*Dy^2+2*es^5*x*Dx-Dx^2+3*es^6*y^2*Dy+3*es^5*y*Dy+es*Dx-2*es^3*Dy*h^2+es^6*y*h^2 , [ Dx , 3*y*Dy-2*h^2 , 0 , 0 ] , 0 , 2 , 2 , 2 ]
276: [ 0 , 4 ]
277: Processing [ 0 , 4 ] Strategy = 5
278: [ 1 , 1 ]
279: Processing [ 1 , 1 ] Strategy = 5
280: [ 2 , 2 ]
281: Processing [ 2 , 2 ] Strategy = 5
282: [ 2 , 3 ]
283: Processing [ 2 , 3 ] Strategy = 5
284: SpairAndReduction:
285: [ p and bases , [ [ 0 , 6 ] , [ -Dx^2 , -3*y^2*Dy ] ] , [ [ 0 , 9*y^2*Dy+12*y*h^2 , 0 , 2*x , 1 ] , -3*es^3*y*Dy+es^4*Dx+2*es^2*x*h^2+9*es*y*Dx*h^2+4*es^3*h^2+6*y^2*Dy*h-4*es*x^2*Dy*h+2*y*h^3-2*es*x*y*h^2 , [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ] , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , %[null] , -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] ] ]
286: [ Dx^2 , -3*es^6*y^2*Dy ]
287: [gi, gj] = [ 9*es*y^2*Dy+2*es^3*x+es^4+12*es*y*h^2 , 3*es*Dx^2-es^2*Dy+3*Dy^2-2*Dx*h-es*h^2 ]
288: 1
289: Reduce the element 3*es^2*y^2*Dy^2+2*es^3*x*Dx^2-9*y^2*Dy^3+es^4*Dx^2+12*es*y*Dx^2*h^2+4*es^3*Dx*h^2+6*y^2*Dx*Dy*h+3*es*y^2*Dy*h^2
290: by [ [ 0 , 9*y^2*Dy+12*y*h^2 , 0 , 2*x , 1 ] , -3*es^3*y*Dy+es^4*Dx+2*es^2*x*h^2+9*es*y*Dx*h^2+4*es^3*h^2+6*y^2*Dy*h-4*es*x^2*Dy*h+2*y*h^3-2*es*x*y*h^2 , [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ] , [ 2*x*Dx , -2*x*h , y , 0 , 0 ] , %[null] , -3*y*Dy^2+es^3*Dx-es^2*h^2+2*Dy*h^2+2*es*x*Dy*h+es*y*h^2 , [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ] ]
291: result is [ 9*es^2*y^2*Dy^2+6*es^3*x*Dx^2-6*es^2*x*Dx*h^2+12*es^2*y*Dy*h^2-6*es^2*h^4+12*es*x^2*Dx*Dy*h-18*es*x*y*Dy^2*h+24*es*x*Dy*h^3+6*es*x*y*Dx*h^2 , 3 , [ 0 , -3*Dx , 0 , 0 , 0 , -9*y*Dy , -3*y*h^2 ] ]
292: vdegree of the original = 2
293: vdegree of the remainder = 2
294: [ 9*es^2*y^2*Dy^2+6*es^3*x*Dx^2-6*es^2*x*Dx*h^2+12*es^2*y*Dy*h^2-6*es^2*h^4+12*es*x^2*Dx*Dy*h-18*es*x*y*Dy^2*h+24*es*x*Dy*h^3+6*es*x*y*Dx*h^2 , [ 3*Dx^2 , -3*Dx , 0 , 0 , 0 , -9*y*Dy , -9*y^2*Dy-3*y*h^2 ] , 3 , 4 , 2 , 2 ]
295: [ 1 , 4 ]
296: Processing [ 1 , 4 ] Strategy = 6
297: [seq,level,q]=[ 6 , 1 , 4 ]
298: [ level, q = , 1 , 4 ]
299: bases=
300: [
301: [ -Dx , 1 , 3*y*Dy-2*h^2 , 0 , 0 , 2*x , 0 ]
302: [ 0 , 0 , -Dx , Dy , 0 , 1 , y ]
303: [ -Dx^2 , Dx , 0 , 3*y*Dy^2-2*Dy*h^2 , 0 , 2*x*Dx+3*y*Dy , 3*y^2*Dy+y*h^2 ]
304: [ -3*Dx^2 , 3*Dx , 0 , 0 , 1 , 9*y*Dy , 9*y^2*Dy+3*y*h^2 ]
305: ]
306: dr=
307: [ 3*Dx^2 , -3*Dx , 0 , 0 , -1 , -9*y*Dy , -9*y^2*Dy-3*y*h^2 ]
308: newbases=
309: [
310: [ -Dx , 1 , 3*y*Dy-2*h^2 , 0 , 0 , 2*x , 0 ]
311: [ 0 , 0 , -Dx , Dy , 0 , 1 , y ]
312: [ -Dx^2 , Dx , 0 , 3*y*Dy^2-2*Dy*h^2 , 0 , 2*x*Dx+3*y*Dy , 3*y^2*Dy+y*h^2 ]
313: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
314: ]
315: [seq,level,q]=[ 5 , 2 , 2 ]
316: [ level, q = , 2 , 2 ]
317: bases=
318: [
319: [ -Dx , -3*y*Dy+2*h^2 , 1 , 0 ]
320: ]
321: dr=
322: [ Dx , 3*y*Dy-2*h^2 , -1 , 0 ]
323: newbases=
324: [
325: [ 0 , 0 , 0 , 0 ]
326: ]
327: [seq,level,q]=[ 4 , 1 , 1 ]
328: [ level, q = , 1 , 1 ]
329: bases=
330: [
331: [ -Dx , 1 , 3*y*Dy-2*h^2 , 0 , 0 , 2*x , 0 ]
332: [ 0 , 0 , -Dx , Dy , 0 , 1 , y ]
333: [ -Dx^2 , Dx , 0 , 3*y*Dy^2-2*Dy*h^2 , 0 , 2*x*Dx+3*y*Dy , 3*y^2*Dy+y*h^2 ]
334: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
335: ]
336: dr=
337: [ Dx , -1 , -3*y*Dy+2*h^2 , 0 , 0 , -2*x , 0 ]
338: newbases=
339: [
340: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
341: [ 0 , 0 , -Dx , Dy , 0 , 1 , y ]
342: [ 0 , 0 , -3*y*Dx*Dy+2*Dx*h^2 , 3*y*Dy^2-2*Dy*h^2 , 0 , 3*y*Dy-2*h^2 , 3*y^2*Dy+y*h^2 ]
343: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
344: ]
345: [seq,level,q]=[ 3 , 0 , 4 ]
346: [ level, q = , 0 , 4 ]
347: bases=
348: [
349: [ 0 , 9*y^2*Dy+12*y*h^2 , 0 , 2*x , 1 ]
350: [ 6*y^2*Dy*h+2*y*h^3 , 9*y*Dx*h^2-4*x^2*Dy*h-2*x*y*h^2 , 2*x*h^2 , -3*y*Dy+4*h^2 , Dx ]
351: [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ]
352: [ 2*x*Dx , -2*x*h , y , 0 , 0 ]
353: [ 0 , 12*x^2*Dx*Dy*h-18*x*y*Dy^2*h+24*x*Dy*h^3+6*x*y*Dx*h^2 , 9*y^2*Dy^2-6*x*Dx*h^2+12*y*Dy*h^2-6*h^4 , 6*x*Dx^2 , 0 ]
354: [ -3*y*Dy^2+2*Dy*h^2 , 2*x*Dy*h+y*h^2 , -h^2 , Dx , 0 ]
355: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ]
356: ]
357: dr=
358: [ 0 , -9*y^2*Dy-12*y*h^2 , 0 , -2*x , -1 ]
359: newbases=
360: [
361: [ 0 , 0 , 0 , 0 , 0 ]
362: [ 6*y^2*Dy*h+2*y*h^3 , -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h-2*x*y*h^2 , 2*x*h^2 , -2*x*Dx-3*y*Dy+2*h^2 , 0 ]
363: [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ]
364: [ 2*x*Dx , -2*x*h , y , 0 , 0 ]
365: [ 0 , 12*x^2*Dx*Dy*h-18*x*y*Dy^2*h+24*x*Dy*h^3+6*x*y*Dx*h^2 , 9*y^2*Dy^2-6*x*Dx*h^2+12*y*Dy*h^2-6*h^4 , 6*x*Dx^2 , 0 ]
366: [ -3*y*Dy^2+2*Dy*h^2 , 2*x*Dy*h+y*h^2 , -h^2 , Dx , 0 ]
367: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ]
368: ]
369: [seq,level,q]=[ 2 , 1 , 5 ]
370: [ level, q = , 1 , 5 ]
371: bases=
372: [
373: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
374: [ 0 , 0 , -Dx , Dy , 0 , 1 , y ]
375: [ 0 , 0 , -3*y*Dx*Dy+2*Dx*h^2 , 3*y*Dy^2-2*Dy*h^2 , 0 , 3*y*Dy-2*h^2 , 3*y^2*Dy+y*h^2 ]
376: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
377: ]
378: dr=
379: [ 0 , 0 , Dx , -Dy , 0 , -1 , -y ]
380: newbases=
381: [
382: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
383: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
384: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
385: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
386: ]
387: [seq,level,q]=[ 1 , 0 , 3 ]
388: [ level, q = , 0 , 3 ]
389: bases=
390: [
391: [ 0 , 0 , 0 , 0 , 0 ]
392: [ 6*y^2*Dy*h+2*y*h^3 , -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h-2*x*y*h^2 , 2*x*h^2 , -2*x*Dx-3*y*Dy+2*h^2 , 0 ]
393: [ 2*x*Dy-2*y*h , 3*y*Dx , 0 , 1 , 0 ]
394: [ 2*x*Dx , -2*x*h , y , 0 , 0 ]
395: [ 0 , 12*x^2*Dx*Dy*h-18*x*y*Dy^2*h+24*x*Dy*h^3+6*x*y*Dx*h^2 , 9*y^2*Dy^2-6*x*Dx*h^2+12*y*Dy*h^2-6*h^4 , 6*x*Dx^2 , 0 ]
396: [ -3*y*Dy^2+2*Dy*h^2 , 2*x*Dy*h+y*h^2 , -h^2 , Dx , 0 ]
397: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ]
398: ]
399: dr=
400: [ -2*x*Dy+2*y*h , -3*y*Dx , 0 , -1 , 0 ]
401: newbases=
402: [
403: [ 0 , 0 , 0 , 0 , 0 ]
404: [ 4*x^2*Dx*Dy+6*x*y*Dy^2-4*x*y*Dx*h , 6*x*y*Dx^2-4*x^2*Dy*h-2*x*y*h^2 , 2*x*h^2 , 0 , 0 ]
405: [ 0 , 0 , 0 , 0 , 0 ]
406: [ 2*x*Dx , -2*x*h , y , 0 , 0 ]
407: [ -12*x^2*Dx^2*Dy-24*x*Dx*Dy*h^2+12*x*y*Dx^2*h , -18*x*y*Dx^3+12*x^2*Dx*Dy*h-18*x*y*Dy^2*h+24*x*Dy*h^3+6*x*y*Dx*h^2 , 9*y^2*Dy^2-6*x*Dx*h^2+12*y*Dy*h^2-6*h^4 , 0 , 0 ]
408: [ -2*x*Dx*Dy-3*y*Dy^2+2*y*Dx*h , -3*y*Dx^2+2*x*Dy*h+y*h^2 , -h^2 , 0 , 0 ]
409: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ]
410: ]
411: [ level= , 0 ]
412: [
413: [ 3*y*Dx^2-2*x*Dy*h-y*h^2 ]
414: [ -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h ]
415: [ -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 ]
416: ]
417: [
418: [ 3*y*Dx^2-2*x*Dy*h-y*h^2 ]
419: [ -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h ]
420: [ -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 ]
421: ]
422: [ level= , 1 ]
423: [
424: [ 0 , 0 , 0 , 0 , 0 ]
425: [ 0 , 0 , 0 , 0 , 0 ]
426: [ 2*x*Dx , -2*x*h , y , 0 , 0 ]
427: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy , 0 , 0 ]
428: ]
429: [
430: [ 0 , 0 , 0 ]
431: [ 0 , 0 , 0 ]
432: [ 2*x*Dx , -2*x*h , y ]
433: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy ]
434: ]
435: [ level= , 2 ]
436: [
437: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
438: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
439: [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ]
440: ]
441: [
442: [ 0 , 0 , 0 , 0 ]
443: [ 0 , 0 , 0 , 0 ]
444: [ 0 , 0 , 0 , 0 ]
445: ]
446: [ level= , 3 ]
447: [
448: [ 0 , 0 , 0 , 0 ]
449: ]
450: [
451: [ 0 , 0 , 0 ]
452: ]
453: In(5)=b=a[0];
454: In(6)=b[1]*b[0]:
455: [ [ 0 ] , [ 0 ] , [ 0 ] , [ 0 ] ]
456: In(7)=b[2]*b[1]:
457: [ [ 0 , 0 , 0 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 0 ] ]
458: In(8)=sm1_pmat(b);
459: [
460: [
461: [ 3*y*Dx^2-2*x*Dy*h-y*h^2 ]
462: [ -2*x*Dx*Dy-3*y*Dy^2-2*Dy*h^2+2*y*Dx*h ]
463: [ -6*x*Dx^3-6*x*Dy^2*h+6*x*Dx*h^2 ]
464: ]
465: [
466: [ 0 , 0 , 0 ]
467: [ 0 , 0 , 0 ]
468: [ 2*x*Dx , -2*x*h , y ]
469: [ 3*Dy^2-2*Dx*h , 3*Dx^2-h^2 , -Dy ]
470: ]
471: [
472: [ 0 , 0 , 0 , 0 ]
473: [ 0 , 0 , 0 , 0 ]
474: [ 0 , 0 , 0 , 0 ]
475: ]
476: [
477: [ 0 , 0 , 0 ]
478: ]
479: ]
480: In(9)=
481:
482:
483: ------- failed example.
484: def Sannfs2_laScala(f) {
485: local p,pp;
486: p = Sannfs(f,"x,y");
487: /* Do not make laplace transform.
488: sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");
489: p = [p]
490: */
491: ;
492:
493: xy(x-y) $B$N(B annihilating ideal $B$N(B V-minimal free resolution.
494: In(6)=a=Sannfs2_laScala("x*y*(x-y)");
495:
496: SpairAndReduction:
497: [ p and bases , [ [ 0 , 1 ] , [ -3*y*Dy , Dx ] ] , [ -4*x*Dx-4*y*Dy-12*h^2 , -12*x*y*Dy+12*y^2*Dy-12*x*h^2+24*y*h^2 , %[null] ] ]
498: [ -3*y*Dy , es*Dx ]
499: [gi, gj] = [ -4*x*Dx-4*y*Dy-12*h^2 , -12*x*y*Dy+12*y^2*Dy-12*x*h^2+24*y*h^2 ]
500: 1
501: Reduce the element 12*y^2*Dx*Dy+12*y^2*Dy^2-12*x*Dx*h^2+24*y*Dx*h^2+36*y*Dy*h^2-12*h^4
502: by [ -4*x*Dx-4*y*Dy-12*h^2 , -12*x*y*Dy+12*y^2*Dy-12*x*h^2+24*y*h^2 , %[null] ]
503: result is [ -12*y^2*Dx*Dy-12*y^2*Dy^2-24*y*Dx*h^2-48*y*Dy*h^2-24*h^4 , -1 , [ 3*h^2 , 0 , 0 ] ]
504: vdegree of the original = 0
505: vdegree of the remainder = 0
506: [ -12*y^2*Dx*Dy-12*y^2*Dy^2-24*y*Dx*h^2-48*y*Dy*h^2-24*h^4 , [ 3*y*Dy+3*h^2 , -Dx , 0 ] , 1 , 2 , 0 , 0 ]
507: reductionTable_tmp=[ 2 ]
508: See also reductionTable, strategy, level,i
509: ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
510: --- Engine error or interrupt : In function : Error of class PrimitiveObject
511:
512: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
513:
514: In(7)=bases:
515: [ %[null] , [ -3*y*Dy-3*h^2 , Dx , 1 ] , %[null] ]
516: In(8)=reductionTable:
517: [ [ 1 , 2 , 3 ] , [ 3 , 2 , 3 ] , [ 2 ] ]
518: In(9)=freeRes:
519: [ [ -4*x*Dx-4*y*Dy-12*h^2 , -12*x*y*Dy+12*y^2*Dy-12*x*h^2+24*y*h^2 , -12*y^2*Dx*Dy-12*y^2*Dy^2-24*y*Dx*h^2-48*y*Dy*h^2-24*h^4 ] , [ %[null] , [ -3*y*Dy-3*h^2 , Dx , 1 ] , %[null] ] , [ %[null] ] ]
520: In(10)=
521:
522: [ [ -4*x*Dx-4*y*Dy-12*h^2 , -12*x*y*Dy+12*y^2*Dy-12*x*h^2+24*y*h^2 , -12*y^2*Dx*Dy-12*y^2*Dy^2-24*y*Dx*h^2-48*y*Dy*h^2-24*h^4 ] ,
523: [ %[null] , [ -3*y*Dy-3*h^2 , Dx , 1 ] , %[null] ] ,
524: $B$3$l$H(B $B$3$l$N(B spair $B$r7W;;$7$h$&$H$7$F;_$^$k(B.
525: [ %[null] ] ]
526:
527: $B$3$l$,(B, strategy $B$N(B table.
528:
529: In(8)=reductionTable:
530: [ [ 1 , 2 , 3 ] , [ 3 , 2 , 3 ] , [ 2 ] ]
531: $B$3$N85$r=hM}Cf(B.
532:
533:
534:
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