Annotation of OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi, Revision 1.7
1.7 ! takayama 1: %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.6 2021/12/11 11:40:45 takayama Exp $
1.1 takayama 2: %% xetex mt_gkz-en.texi (.texi までつける. )
3: %% @math{tex形式の数式}
4: %% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183
5: %% @tex{tex形式で書いたもの}
6: %%https://www.gnu.org/software/texinfo/manual/texinfo/html_node/_0040TeX-_0040LaTeX.html
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8: %% 英語版, 以下コメントは @comment で始める. \input texinfo 以降は普通の tex 命令は使えない.
9: \input texinfo-ja
10: @iftex
11: @catcode`@#=6
12: @def@fref#1{@xrefX[#1,,@code{#1},,,]}
13: @def@b#1{{@bf #1}}
14: @catcode`@#=@other
15: @end iftex
16: @overfullrule=0pt
17: @documentlanguage en
18: @c -*-texinfo-*-
19: @comment --- おまじない終り ---
20:
21: @comment --- GNU info ファイルの名前 ---
1.4 takayama 22: @setfilename asir-contrib-mt_gkz_man
1.1 takayama 23:
24: @comment --- タイトル ---
25: @settitle GKZ hypergeometric system
26:
27: @comment --- おまじない ---
28: @ifinfo
29: @macro fref{name}
30: @ref{\name\,,@code{\name\}}
31: @end macro
32: @end ifinfo
33:
34: @titlepage
35: @comment --- おまじない終り ---
36:
37: @comment --- タイトル, バージョン, 著者名, 著作権表示 ---
38: @title GKZ hypergeometric system
39: @subtitle Pfaffian system (Pfaff equation), contiguity relations, cohomology intersection
40: @subtitle Version 1.0
1.6 takayama 41: @subtitle December 21, 2021
1.1 takayama 42:
43: @author by S-J. Matsubara-Heo, N.Takayama
44: @page
45: @vskip 0pt plus 1filll
46: Copyright @copyright{} Risa/Asir committers
47: 2004--2020. All rights reserved.
48: @end titlepage
49:
50: @comment --- おまじない ---
51: @synindex vr fn
52: @comment --- おまじない終り ---
53:
54: @comment --- @node は GNU info, HTML 用 ---
55: @comment --- @node の引数は node-name, next, previous, up ---
56: @node Top,, (dir), (dir)
57:
58: @comment --- @menu は GNU info, HTML 用 ---
59: @comment --- chapter 名を正確に並べる ---
60:
61: @menu
62: * About this document::
63: * Pfaff equation::
64: * b function::
65: * Utilities::
66: * Index::
67: @end menu
68:
69: @comment --- chapter の開始 ---
70: @comment --- 親 chapter 名を正確に. 親がない場合は Top ---
71: @node About this document,,, Top
72: @chapter About this document
73:
74: This document explains Risa/Asir functions for GKZ hypergeometric system
75: (A-hypergeometric system). @* @comment 強制改行
76: Loading the package:
77: @example
78: import("mt_gkz.rr");
79: @end example
80: @noindent
81: References cited in this document.
82: @itemize @bullet
83: @item [MT2020]
84: Saiei-Jaeyeong Matsubara-Heo, Nobuki Takayama,
85: Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals,
86: Lecture Notes in Computer Science 12097 (2020), 73--84.
87: Errata is posted on @uref{http://arxiv.org/abs/???}.
88: E-attachments can be obtainable at
89: @uref{http://www.math.kobe-u.ac.jp/OpenXM/Math/intersection2}
90: @item [GM2020]
91: Yoshiaki Goto, Saiei-Jaeyeong Matsubara-Heo,
92: Homology and cohomology intersection numbers of GKZ systems, arXiv:2006.07848
93: @item [SST1999]
94: M.Saito, B.Sturmfels, N.Takayama, Hypergeometric polynomials
95: and integer programming, Compositio Mathematica, 155 (1999), 185--204
96: @item [SST2000]
97: M.Saito, B.Sturmfels, N.Takayama, Groebner Deformations of Hypergeometric
98: Differential Equations. Springer, 2000.
99: @end itemize
100:
101: References for maple packages IntegrableConnections and OreMorphisms.
102: @itemize @bullet
103: @item [BCEW]
104: M.Barkatou, T.Cluzeau, C.El Bacha, J.-A.Weil,
105: IntegrableConnections – a maple package for computing closed form solutions of integrable connections
106: (2012). @uref{https://www.unilim.fr/pages perso/thomas.cluzeau/Packages/IntegrableConnections/PDS.html}
107: @item [CQ]
108: T.Cluzeau and A.Quadrat,
109: OreMorphisms: A homological algebraic package for factoring, reducing and decomposing linear functional systems (2009). @uref{https://who.rocq.inria.fr/Alban.Quadrat/OreMorphisms/index.html}
110: @item [CQ08]
111: T.Cluzeau, A.Quadrat, Factoring and decomposing a class of linear functional
112: systems, Linear Algebra and its Applications (LAA), 428(1): 324-381, 2008.
113: @end itemize
114:
115:
116:
117: @node Pfaff equation,,, Top
118: @chapter Pfaff equation
119:
120: @menu
121: * mt_gkz.pfaff_eq::
122: * mt_gkz.ff::
123: * mt_gkz.ff1::
124: * mt_gkz.ff2::
125: * mt_gkz.rvec_to_fvec::
1.6 takayama 126: * mt_gkz.fvec_to_conn_mat::
127: * mt_gkz.contiguity::
1.1 takayama 128: @end menu
129:
130: @node Pfaff equation for given cocycles,,, Pfaff equation
131: @section Pfaff equation for given cocycles
132:
133: @comment **********************************************************
134: @comment --- 関数 pfaff_eq
135: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
136: @subsection @code{mt_gkz.pfaff_eq}
137: @comment --- 索引用キーワード
138: @findex mt_gkz.pfaff_eq
1.5 takayama 139: @findex mt_gkz.use_hilbert_driven
1.1 takayama 140:
141: @table @t
142: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
143: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
144: @end table
145:
146: @comment --- 引数の簡単な説明 ---
147: @table @var
148: @item return
149: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
150: @item A
151: the matrix A of the GKZ system.
152: @item Beta
153: the parameter vector of the GKZ system.
154: @item Ap
155: See [MT2020].
156: @item Rvec
1.6 takayama 157: It is used to specify a basis of cocycles as explained below. See also [MT2020].
1.1 takayama 158: @item DirX
159: a list of dxi's.
160: @end table
161:
162: @comment --- ここで関数の詳しい説明 ---
163: @comment --- @itemize〜@end itemize は箇条書き ---
164: @comment --- @bullet は黒点付き ---
165: @itemize @bullet
166: @item
167: The independent variables are x1, x2, x3, ...
168: @item
169: When @var{Rvec}=[v_1, v_2, ..., v_r] where r is the rank of the GKZ system,
170: the set of the cocycles standing for Av_1, Av_2, ..., Av_r
171: (see [MT2020])
172: is supposed
173: to be the basis to construct the Pfaffian system.
1.6 takayama 174: The exponents @math{(q_\ell, q)} of the integral representation
175: @math{\int \prod h_\ell^{-q_\ell} x^q {{dx} \over {x}}}
176: is shifted by Av_i@math{=:A_{v_i}} as @math{(q_\ell,q)+A_{v_i}}.
1.1 takayama 177: Let a_1, a_2, ..., a_n be the column vectors of the matrix A
178: and v be a column vector (x_1, x_2, ..., x_n)^T.
179: Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n.
180: @item
181: When the columns of @var{A} are expressed as
182: @math{e_i \otimes \alpha_{i_j}},
183: the columns of @var{Ap} is
184: @math{e_i \otimes 0} where $e_i$ is the i-th unit vector.
185: See [MT2020] on the definition of @var{Ap}.
186: Here are some examples.
187: When @var{A} is
188: @verbatim
189: [[1,1,0,0],
190: [0,0,1,1],
191: [0,1,0,1]]
192: @end verbatim
193: @var{Ap} is
194: @verbatim
195: [[1,1,0,0],
196: [0,0,1,1],
197: [0,0,0,0]] <-- zero row
198: @end verbatim
199: When @var{A} is
200: @verbatim
201: [[1,1,1,0,0,0],
202: [0,0,0,1,1,1],
203: [0,1,0,0,1,0],
204: [0,0,1,0,0,1]
205: ]
206: @end verbatim
207: @var{Ap} is
208: @verbatim
209: [[1,1,1,0,0,0],
210: [0,0,0,1,1,1],
211: [0,0,0,0,0,0], <-- zero row
212: [0,0,0,0,0,0] <-- zero row
213: ]
214: @end verbatim
215: See also page 223 of [SST2000].
216: @item
217: Option @var{xrule}. When the option @var{xrule} is given,
218: the x variables specified by this option are specialized to numbers.
219: @item
220: Option @var{shift}. When the matrix @var{A} is not normal
221: (the associated toric ideal is not normal), a proper shift vector
222: must be given to obtain an element of the b-ideal. Or, use the option
223: @var{b_ideal} below. See [SST1999] on the theory.
224: @item
225: Option @var{b_ideal}. When the matrix @var{A} is not normal,
226: the option @code{b_ideal=1} obtains b-ideals and the first element
227: of each b-ideal is used as the b-function. The option @var{shift}
228: is ignored.
229: @item
230: Option @var{cg}. A constant matrix given by this option is used
231: for the Gauge transformation of the Pfaffian system.
232: In other words, the basis of cocycles specified by @var{Rvec}
233: is transformed by the constant matrix given by this option.
1.5 takayama 234: @item
235: By mt_gkz.use_hilbert_driven(Rank), the rank of the GKZ system is assumed to be
236: Rank. It makes the computation of Groebner basis by yang.rr faster.
237: This option is disabled by mt_gkz.use_hilbert_driven(0);
1.1 takayama 238: @end itemize
239:
240: @comment --- @example〜@end example は実行例の表示 ---
241: Example: Gauss hypergeometric system, see [GM2020] example ??.
242: @example
243: [1883] import("mt_gkz.rr");
244: [2657] PP=mt_gkz.pfaff_eq(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
245: Beta=[-g1,-g2,-c],
246: Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
247: Rvec = [[1,0,0,0],[0,0,1,0]],
248: DirX=[dx4,dx3] | xrule=[[x1,1],[x2,1]],
249: cg=matrix_list_to_matrix([[1,0],[-1,1]]))$
250:
251: Bfunctions=[s_1*s_2-s_1*s_3+s_1^2,s_1*s_3,s_2^2+(-s_3+s_1)*s_2,s_3*s_2]
252: -- snip --
253: [2658] PP[0];
254: [ (g2*x3-g2)/(x4-x3) (g2*x3)/(x4-x3) ]
255: [ ((-g2*x3-c+g2)*x4+(c-g1)*x3+g1)/(x4^2-x3*x4)
256: ((-g2*x3-c)*x4+(c-g1)*x3)/(x4^2-x3*x4) ]
257: [2659] PP[1];
258: [ (-g2*x4+g2)/(x4-x3) (-g2*x4)/(x4-x3) ]
259: [ ((g2*x3+c-g2-1)*x4+(-c+g1+1)*x3-g1)/(x3*x4-x3^2)
260: ((g2*x3+c-g2-1)*x4+(-c+g1+g2+1)*x3)/(x3*x4-x3^2) ]
261: @end example
262:
263: @*
264:
265: Example: The role of shift.
266: When the toric ideal is not normal, a proper shift vector
267: must be given with the option @code{shift} to find an element of the b-ideal.
268: @example
269: [1882] load("mt_gkz.rr");
270: [1883] A=[[1,1,1,1],[0,1,3,4]];
271: [[1,1,1,1],[0,1,3,4]]
272: [1884] Ap=[[1,1,1,1],[0,0,0,0]];
273: [[1,1,1,1],[0,0,0,0]]
274: [1885] Rvec=[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
275: [[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
276: [2674] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
277: | xrule=[[x1,1],[x2,2],[x3,4]] )$
278: dx remains
279: stopped in step_up at line 342 in file "./mt_gkz/saito-b.rr"
280: 342 if (type(dn(Ans)) > 1) error("dx remains");
281: (debug) quit
282: // Since the toric ideal for A is not normal, it stops with the error.
283: [2675] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
284: | shift=[1,0],xrule=[[x1,1],[x2,2],[x3,4]])$
285: // It works.
286: @end example
287:
288: @comment --- 参照(リンク)を書く ---
289: @table @t
290: @item Refer to
291: @ref{mt_gkz.ff1}
292: @ref{mt_gkz.ff2}
293: @ref{mt_gkz.ff}
294: @ref{mt_gkz.rvec_to_fvec}
295: @end table
296: @comment mt_gkz.pfaff_eq の説明おわり. あとはこれの繰り返し.
297:
298:
299: @comment --- 個々の関数の説明 ---
300: @comment --- section 名を正確に ---
301: @node mt_gkz.ff2,,, Pfaff equation for given cocycles
302: @node mt_gkz.ff1,,, Pfaff equation for given cocycles
303: @node mt_gkz.ff,,, Pfaff equation for given cocycles
304: @subsection @code{mt_gkz.ff2}, @code{mt_gkz.ff1}, @code{mt_gkz.ff}
305: @comment --- 索引用キーワード
306: @findex mt_gkz.ff2
307: @findex mt_gkz.ff1
308: @findex mt_gkz.ff
309:
310: @table @t
1.7 ! takayama 311: @item mt_gkz.ff(@var{Rvec0},@var{A},@var{Ap},@var{Beta})
1.1 takayama 312: @item mt_gkz.ff1(@var{Rvec0},@var{A},@var{Beta},@var{Ap})
313: @item mt_gkz.ff2(@var{Rvec0},@var{A},@var{Beta},@var{Ap},@var{BF},@var{C})
314: :: @code{ff} returns a differential operator whose action to 1 gives
315: the cocycle defined by @var{Rvec0}
316: @end table
317:
318: @comment --- 引数の簡単な説明 ---
319: @table @var
320: @item return
321: @code{ff} returns a differential operator whose action to 1 of @math{M_A(\beta)}
322: gives the cocycle defined by @var{Rvec0}.
323: @item return
324: @code{ff1} returns a composite of step-down operators for the positive part
325: of @var{Rvec0}
326: @item return
327: @code{ff2} returns a composite of step-up operators for the positive part
328: of @var{Rvec0}
329: @item Rvec0
330: An element of @var{Rvec} explained in @ref{mt_gkz.pfaff_eq}.
331: @item BF
332: the list of b-functions to all directions.
333: @item C
334: the list of the step up operators for all a_1, a_2, ..., a_n.
335: @end table
336: Other arguments are same with those of @code{pfaff_eq}.
337:
338: @comment --- ここで関数の詳しい説明 ---
339: @comment --- @itemize〜@end itemize は箇条書き ---
340: @comment --- @bullet は黒点付き ---
341: @itemize @bullet
342: @item
343: The function @code{ff} generates the list of b-functions and the list of
344: step up operators and store them in the cache variable.
345: They can be obtained by calling as @code{S=mt_gkz.get_bf_step_up()}
346: where S[0] is the list of b-functions and S[1] is the list of step up
347: operators.
348: Step up operators are obtained by the algorithm given in [SST1999].
349: @item
350: Option nf. When nf=1, the output operator is reduced to the normal form
351: with respect to the Groebner basis of the GKZ system of the graded reverse
352: lexicographic order.
353: @item
354: Option shift. See @ref{mt_gkz.pfaff_eq}.
355: @item
356: Internal info: The function @code{mt_gkz.bb} gives the constant so that
357: the step up and step down operators (contiguity operators) give
358: contiguity relations for the integral representation in [MT2020].
359: Note that @code{mt_gkz.ff1} and @code{mt_gkz.ff2} give contiguity
360: relations which are constant multiple of those for hypergeometric
361: polynomials.
362: @item
363: Internal info: @code{mt_gkz.step_up} generates step up operators
364: of [SST1999] from b-functions by utilizing @code{mt_gkz.bf2euler}
365: and @code{mt_gkz.toric}.
366: @end itemize
367:
368: @comment --- @example〜@end example は実行例の表示 ---
369: Example: Step up operators compatible with the integral representation in [MT2020].
370: The function hgpoly_res defined in @code{check-by-hgpoly.rr} returns
371: a multiple of the hypergeometric polynomial which agrees with
372: the residue times a power of @math{2\pi \sqrt{-1}}
373: of the integral representation.
374: See [SST1999].
375: @example
376: [1883] import("mt_gkz.rr")$
377: [3175] load("mt_gkz/check-by-hgpoly.rr")$
378: [3176] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
379: [3177] B=newvect(3,[5,4,7])$ Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
380: [3179] Beta=[b1,b2,b3]$ R=[0,0,-1,0]$
381: [3180] F2=hgpoly_res(A,B,2); // HG polynomial. 2 is the number of e_i's.
382: 10*x1^2*x2^3*x4^4+20*x1*x2^4*x3*x4^3+6*x2^5*x3^2*x4^2
383: [3182] mt_gkz.ff(R,A,Ap,Beta); // the operator standing for R
384: (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
385: [3184] S=mt_gkz.get_bf_step_up(A); // b-function and non-reduced step up op's
386: [[ s_1*s_2-s_1*s_3+s_1^2 s_1*s_3 s_2^2+(-s_3+s_1)*s_2 s_3*s_2 ],
387: [ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1
388: x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2
389: x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3
390: x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]
391: [3185] Fvec=mt_gkz.ff2(R,A,Beta,Ap,S[0],S[1]);
392: (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
393: [3188] Fvec = base_replace(Fvec,assoc(Beta,vtol(B)));
394: 1/3*x3*x4*dx4+1/3*x3^2*dx3+1/3*x1*x4*dx2+1/3*x1*x3*dx1+1/3*x3
395: [3189] R32d = odiff_act(Fvec,F2,[x1,x2,x3,x4]); // Act Fvec to the hg-poly
396: 10*x1^3*x2^2*x4^5+50*x1^2*x2^3*x3*x4^4+50*x1*x2^4*x3^2*x4^3+10*x2^5*x3^3*x4^2
397: [3190] red(R32d/hgpoly_res(A,B+newvect(3,[0,1,0]),2));
398: // R32d agrees with the HG polynomial with Beta=[5,4,7]+[0,1,0].
399: 1
400: @end example
401:
402: @comment --- 参照(リンク)を書く ---
403: @table @t
404: @item Refer to
405: @ref{mt_gkz.pfaff_eq}
406: @end table
407: @comment おわり.
408:
409: @comment --- 個々の関数の説明 --- Ref:2020-11-09-tw-cohom-progs.goodnotes
410: @comment --- section 名を正確に ---
411: @node mt_gkz.rvec_to_fvec,,, Pfaff equation for given cocycles
412: @subsection @code{mt_gkz.rvec_to_fvec}
413: @comment --- 索引用キーワード
414: @findex mt_gkz.rvec_to_fvec
415:
416: @table @t
417: @item mt_gkz.rvec_to_fvec(@var{Rvec},@var{A},@var{Ap},@var{Beta})
418: :: It returns a set of differential operators standing for @var{Rvec}.
419: @end table
420:
421: @comment --- 引数の簡単な説明 ---
422: @table @var
423: @item return
424: It returns a set of differential operators of which action to
425: @math{1 \in M_A(\beta)} give cocycles specified by @var{Rvec}.
426: @item A, Ap, Beta
427: Same with @ref{mt_gkz.pfaff_eq}
428: @end table
429:
430: @comment --- ここで関数の詳しい説明 ---
431: @comment --- @itemize〜@end itemize は箇条書き ---
432: @comment --- @bullet は黒点付き ---
433: @itemize @bullet
434: @item
435: Internal info: this function builds the set of operators by calling
436: @ref{mt_gkz.ff}.
437: @end itemize
438:
439: @comment --- @example〜@end example は実行例の表示 ---
440: Example: The following two expressions are congruent because
441: @math{2a_1-a_2-a_3+a_4=a_1} for this @code{A}.
442: @example
443: [1883] import("mt_gkz.rr");
444: [3191] mt_gkz.rvec_to_fvec([[2,-1,-1,1],[0,0,1,0]],
445: [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
446: [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
447: [(x2*x3*x4^2*dx1^2*dx4^3+((x1*x3*x4^2+x2*x3^2*x4)*dx1^2*dx3
448: +(x1*x2*x4^2+x2^2*x3*x4)*dx1^2*dx2+(x1^2*x4^2+2*x1*x2*x3*x4+x2^2*x3^2)*dx1^3
449: +(x1*x4^2+3*x2*x3*x4)*dx1^2)*dx4^2+(x1*x3^2*x4*dx1^2*dx3^2
450: +((x1^2*x3*x4+x1*x2*x3^2)*dx1^3+(3*x1*x3*x4+x2*x3^2)*dx1^2)*dx3
451: +x1*x2^2*x4*dx1^2*dx2^2+((x1^2*x2*x4+x1*x2^2*x3)*dx1^3
452: +(3*x1*x2*x4+x2^2*x3)*dx1^2)*dx2+x1^2*x2*x3*dx1^4
453: +(x1^2*x4+3*x1*x2*x3)*dx1^3+(x1*x4+x2*x3)*dx1^2)*dx4)
454: /(b3*b2*b1^3+(b3*b2^2+(-b3^2-2*b3)*b2)*b1^2+(-b3*b2^2+(b3^2+b3)*b2)*b1),
455: (dx3)/(b2)]
456: [3192] mt_gkz.rvec_to_fvec([[1,0,0,0],[0,0,1,0]],
457: [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
458: [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
459: [(dx1)/(b1),(dx3)/(b2)]
460: @end example
461:
462: @comment --- 参照(リンク)を書く ---
463: @table @t
464: @item Refer to
465: @ref{mt_gkz.pfaff_eq}
466: @end table
467: @comment おわり.
468:
469: @comment --- fvec_to_conn_mat
470: @comment --- section 名を正確に ---
471: @node mt_gkz.fvec_to_conn_mat,,, Pfaff equation for given cocycles
472: @subsection @code{mt_gkz.fvec_to_conn_mat}
473: @comment --- 索引用キーワード
474: @findex mt_gkz.fvec_to_conn_mat
475:
476: @table @t
477: @item mt_gkz.fvec_to_conn_mat(@var{Fvec},@var{A},@var{Beta},@var{DirX})
478: :: It returns the coefficient matrices of the basis
479: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis.
480: @end table
481:
482: @comment --- 引数の簡単な説明 ---
483: @table @var
484: @item return
485: It returns the coefficient matrices of the basis
486: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis of the Groebner basis explained below.
487: @item A Beta
488: Same with @ref{mt_gkz.pfaff_eq}.
489: @item DirX
490: When @var{DirX} is 1, this function returns the matrix which expresses
491: @var{Fvec} in terms of the set of the standard monomials of
492: the Groebner basis of the GKZ system in the ring of rational function
493: coefficients with respect to the graded reverse lexicographic order.
494: In other cases, it returns the coefficient matrices of
495: @var{DirX}[I]'s*@var{Fvec} in terms of the set of the standard basis of the Groebner basis.
496: @end table
497:
498: @comment --- ここで関数の詳しい説明 ---
499: @comment --- @itemize〜@end itemize は箇条書き ---
500: @comment --- @bullet は黒点付き ---
501: @itemize @bullet
502: @item
503: It utilizes a Groebner basis computation by the package @code{yang.rr}
504: and @code{yang.reduction} to obtain connection matrices.
505: @item
506: This function calls some utility functions
507: @code{mt_gkz.dmul(Op1,Op2,XvarList)} (multiplication of @code{Op1} and @code{Op2}
508: and @code{mt_gkz.index_vars(x,Start,End | no_=1)}
509: which generates indexed variables without the underbar ``_''.
510: @item
511: We note here some other utility functions in this section:
512: @code{mt_gkz.check_compatibility(P,Q,X,Y)},
513: which checkes if the sytem d/dX-P, d/dY-Q is compatible.
514: @end itemize
515:
516: @comment --- @example〜@end example は実行例の表示 ---
517: Example: The following example illustrates how mt_gkz.pfaff_eq
518: obtains connection matrices.
519: @example
520: [1883] import("mt_gkz.rr");
521: [3201] V=mt_gkz.index_vars(x,1,4 | no_=1);
522: [x1,x2,x3,x4]
523: [3202] mt_gkz.dmul(dx1,x1^2,V);
524: x1^2*dx1+2*x1
525: [3204] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
526: Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
527: Beta= [b1,b2,b3]$
528: Rvec = [[1,0,0,0],[0,0,1,0]]$
529: Fvec = mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)$
530: /* Express cocyles Rvec
531: by elements Fvec in the Weyl algebra by contiguity relations. */
532: Cg = matrix_list_to_matrix([[1,0],[1,-1]])$
533: [3208] NN=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,1);
534: // Express Fvec by the standard monomials Std=NN[1].
535: 1 ooo 2 .ooo
536: [[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
537: [ (-x4)/(b1*x2) (1)/(x3) ],[dx4,1]]
538: [3209] Std=NN[1];
539: [dx4,1]
540: [3173] NN=NN[0];
541: [ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
542: [ (-x4)/(b2*x3) (1)/(x3) ]
543: [3174] NN1=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,dx1)[0];
544: // Express dx1*Fvec by the standard monomials Std.
545: 1 ooo 2 .ooo
546: [ ((2*b1+b2-b3-1)*x1*x4^2+(-b1+b3+1)*x2*x3*x4)/(b1*x1^3*x4-b1*x1^2*x2*x3)
547: ((b1^2+(-2*b3-1)*b1-b3*b2+b3^2+b3)*x1*x4
548: +(-b1^2+(2*b3+1)*b1-b3^2-b3)*x2*x3)/(b1*x1^3*x4-b1*x1^2*x2*x3) ]
549: [(b1 (-b1*x1*x4^2-b2*x2*x3*x4)/(b2*x1^2*x3*x4-b2*x1*x2*x3^2)
550: (b1*x1*x4+(-b1+b3)*x2*x3)/(x1^2*x3*x4-x1*x2*x3^2) ]
551: [3188] P1=map(red,Cg*NN1*matrix_inverse(NN)*matrix_inverse(Cg));
552: [ ((-b2*x3+(b1+b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3)/(x1^2*x4-x1*x2*x3)
553: (b2*x3*x4)/(x1^2*x4-x1*x2*x3) ]
554: [ ((-b2*x3+(b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3+b1*x1*x2)/(x1^2*x4-x1*x2*x3)
555: ((b2*x3+b1*x1)*x4)/(x1^2*x4-x1*x2*x3) ]
556:
557: [3191] mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,[dx1]|cg=Cg)[0]-P1;
558: [ 0 0 ]
559: [ 0 0 ] // P1 agrees with the output of mt_gkz.pfaff_eq.
560: @end example
561:
562: @comment --- 参照(リンク)を書く ---
563: @table @t
564: @item Refer to
565: @ref{mt_gkz.pfaff_eq}
566: @end table
567: @comment おわり.
568:
1.6 takayama 569: @comment --- contiguity
570: @comment --- section 名を正確に ---
571: @node mt_gkz.contiguity,,, Pfaff equation for given cocycles
572: @subsection @code{mt_gkz.contiguity}
573: @comment --- 索引用キーワード
574: @findex mt_gkz.contiguity
575:
576: @table @t
577: @item mt_gkz.contiguity(@var{A},@var{Beta},@var{Ap},@var{Rvec1},@var{Rvec2})
578: :: It returns the coefficient matrix P that satisfies
579: @var{Rvec1} = P @var{Rvec2}.
580: @end table
581:
582: @comment --- 引数の簡単な説明 ---
583: @table @var
584: @item return
585: The coefficient matrix P that satisfies @var{Rvec1} = P @var{Rvec2}.
586: @item A Beta Ap Rvec1 Rvec2
587: Same with @ref{mt_gkz.pfaff_eq}.
588: @end table
589:
590: @comment --- ここで関数の詳しい説明 ---
591: @comment --- @itemize〜@end itemize は箇条書き ---
592: @comment --- @bullet は黒点付き ---
593: @itemize @bullet
594: @item
595: It returns the contiguity relation between
596: @var{Rvec1} and @var{Rvec2}
597: @end itemize
598:
599: @comment --- @example〜@end example は実行例の表示 ---
600: Example:
601: @example
602: [1883] import("mt_gkz.rr");
603: [3200] PP=mt_gkz.contiguity(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
604: Beta=[-g1,-g2,-c],
605: Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
606: Rvec1 = [[1,0,0,0],[0,0,1,0]],
607: Rvec2 = [[0,0,1,0],[1,0,0,0]]);
608: @end example
609:
610: @comment --- 参照(リンク)を書く ---
611: @table @t
612: @item Refer to
613: @ref{mt_gkz.pfaff_eq}
614: @ref{mt_gkz.fvec_to_conn_mat}
615: @end table
616: @comment おわり.
617:
1.1 takayama 618: @comment ---------- New Chapter ---------------
619: @node b function,,, Top
620: @chapter b function
621:
622: @menu
623: * mt_gkz.bf::
624: * mt_gkz.bf::
625: @end menu
626:
627: @node b function and facet polynomial,,, b function
628: @section b function and facet polynomial
629: @comment ------- bf
630: @comment **********************************************************
631: @comment --- 個々の関数の説明 ---
632: @comment --- section 名を正確に ---
633: @node mt_gkz.bf,,, b function and facet polynomial
634: @subsection @code{mt_gkz.bf}
635: @comment --- 索引用キーワード
636: @findex mt_gkz.bf
637:
638: @table @t
639: @item mt_gkz.bf(@var{A},@var{Facet_poly},@var{II0})
640: :: It returns the b-function with respect to the direction @var{II0}.
641: @end table
642:
643: @comment --- 引数の簡単な説明 ---
644: @table @var
645: @item return
646: It returns the b-function introduced Saito with respect to the direction @var{II0} in case of @var{A} is normal or an element of b-ideal when a proper shift vector is given in case of @var{A} is not normal.
647: @item A
648: the matrix A of the GKZ system.
649: @item Facet_poly
650: The set of facet polynomials of the convex hull of @var{A}.
651: @item II0
652: Direction expressed as 0, 1, 2, ... (not 1, 2, 3, ...) to obtain the b function.
653: @end table
654:
655: @comment --- ここで関数の詳しい説明 ---
656: @comment --- @itemize〜@end itemize は箇条書き ---
657: @comment --- @bullet は黒点付き ---
658: @itemize @bullet
659: @item
660: See [SST1999] on the b-function introduced Saito and b-ideal.
661: @item
662: The facet polynomial must be primitive.
663: @end itemize
664:
665: @comment --- @example〜@end example は実行例の表示 ---
666: Example:
667: @example
668: [1883] import("mt_gkz.rr");
669:
670: [3193] A;
671: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
672: [3194] Fpoly=mt_gkz.facet_poly(A);
673: [[s_3,s_1,s_2-s_3+s_1,s_2],[[0,0,1],[1,0,0],[1,1,-1],[0,1,0]]]
674: [3196] mt_gkz.bf(A,Fpoly,0);
675: s_1*s_2-s_1*s_3+s_1^2
676: [3197] mt_gkz.bf(A,Fpoly,1);
677: s_1*s_3
678: @end example
679:
680: @comment --- 参照(リンク)を書く ---
681: @table @t
682: @item Refer to
683: @ref{mt_gkz.ff}
684: @ref{mt_gkz.facet_poly}
685: @end table
686: @comment おわり.
687:
688: @comment ------ facet_poly
689: @comment --- 個々の関数の説明 ---
690: @comment --- section 名を正確に ---
691: @node mt_gkz.facet_polyl,,, b function and facet polynomial
692: @subsection @code{mt_gkz.facet_poly}
693: @comment --- 索引用キーワード
694: @findex mt_gkz.facet_poly
695:
696: @table @t
697: @item mt_gkz.facet_poly(@var{A})
698: :: It returns the set of facet polynomials and their normal vectors of
699: the cone defined by @var{A}.
700: @end table
701:
702: @comment --- 引数の簡単な説明 ---
703: @table @var
704: @item return
705: It returns the set of facet polynomials and their normal vectors of
706: the cone generated by the column vectors of the matrix @var{A}.
707: @item A
708: the matrix A of the GKZ system.
709: @end table
710:
711: @comment --- ここで関数の詳しい説明 ---
712: @comment --- @itemize〜@end itemize は箇条書き ---
713: @comment --- @bullet は黒点付き ---
714: @itemize @bullet
715: @item
716: The facet polynomial f is primitive. In other words,
717: all f(a_i) is integer and min f(a_i)=1 for a_i's not being on f=0.
718: where a_i is the i-th column vector of the matrix @var{A}.
719: It can be checked by @code{mt_gkz.is_primitive(At,Facets)}
720: where @var{At} is the transpose of @var{A} and
721: @var{Facets} is the second return value of this function.
722: @item
723: This function utilizes the system polymake @uref{https://polymake.org}
724: on our server.
725: @end itemize
726:
727: @comment --- @example〜@end example は実行例の表示 ---
728: Example:
729: @example
730: [1883] import("mt_gkz.rr");
731: [1884] mt_gkz.facet_poly([[1,1,1,1],[0,1,2,3]]);
732: oohg_native=0, oohg_curl=1
733: [[s_2,-s_2+3*s_1],[[0,1],[3,-1]]]
734: @end example
735:
736: @comment --- 参照(リンク)を書く ---
737: @table @t
738: @item Refer to
739: @ref{mt_gkz.bf}
740: @end table
741: @comment おわり.
742:
743: @comment ---------- New Chapter ---------------
744: @node utilities,,, Top
745: @chapter Utilities
746:
747: @menu
748: * mt_gkz.reduce_by_toric::
749: * mt_gkz.tk_base_equal::
750: * mt_gkz.dp_op_to_coef_vec::
751: * mt_gkz.yang_gkz_buch::
752: * mt_gkz.p_true_nf_rat::
753: * mt_gkz.mdiff::
754: * mt_gkz.dvar::
755: * mt_gkz.ord_xi::
756: * mt_gkz.get_check_fvec::
757: * mt_gkz.get_bf_step_up::
1.2 takayama 758: * mt_gkz.mytoric_ideal::
1.5 takayama 759: * mt_gkz.cbase_by_euler::
1.1 takayama 760: @end menu
761:
762: @node some utility functions,,, utilities
763: @section Some utility functions
764:
765: @node mt_gkz.reduce_by_toric,,, some utility functions
766: @node mt_gkz.tk_base_equal,,, some utility functions
767: @node mt_gkz.dp_op_to_coef_vec,,, some utility functions
768: @node mt_gkz.yang_gkz_buch,,, some utility functions
769: @node mt_gkz.p_true_nf_rat,,, some utility functions
770: @node mt_gkz.mdiff,,, some utility functions
771: @node mt_gkz.dvar,,, some utility functions
772: @node mt_gkz.ord_xi,,, some utility functions
773: @node mt_gkz.get_check_fvec,,, some utility functions
774: @node mt_gkz.get_bf_step_up,,, some utility functions
1.2 takayama 775: @node mt_gkz.mytoric_ideal,,, some utility functions
1.5 takayama 776: @node mt_gkz.cbase_by_euler,,, some utility functions
1.1 takayama 777:
778: @findex mt_gkz.reduce_by_toric
779: @findex mt_gkz.tk_base_equal
780: @findex mt_gkz.dp_op_to_coef_vec
781: @findex mt_gkz.yang_gkz_buch
782: @findex mt_gkz.p_true_nf_rat
783: @findex mt_gkz.mdiff
784: @findex mt_gkz.dvar
785: @findex mt_gkz.ord_xi
786: @findex mt_gkz.get_check_fvec
787: @findex mt_gkz.get_bf_step_up
1.2 takayama 788: @findex mt_gkz.mytoric_ideal
1.5 takayama 789: @findex mt_gkz.cbase_by_euler
1.1 takayama 790:
791: @comment --- @example〜@end example は実行例の表示 ---
792: We only show examples on these functions. As for details, please see
793: the source code.
794: @example
795: [1883] import("mt_gkz.rr");
796: [2667] mt_gkz.dvar([x1,x2]); // it generates variables starting with d
797: [dx1,dx2]
798: [2669] mt_gkz.p_true_nf_rat((1/3)*x^3-1,[x^2-1],[x],0);
799: [x-3,3] // p_true_nf does not accept rational number coefficients
800: [2670] A=[[1,1,1,1],[0,1,3,4]];
801: [[1,1,1,1],[0,1,3,4]]
802: [2671] mt_gkz.reduce_by_toric(dx3^4,A);
803: dx1*dx4^3 // reduction by toric ideal defined by A
804: [2672] nk_toric.toric_ideal(A);
805: [-x1*x4+x2*x3,-x2*x4^2+x3^3,x2^2*x4-x1*x3^2,-x1^2*x3+x2^3]
806: [2673] mt_gkz.yang_gkz_buch(A,[b1,b2]); // Groebner basis of GKZ system by yang.rr
807: 1 o 2 ..o 3 ..oooooooo 4 o 6 ooo 9 o
808: [[[(x2)*<<0,1,0,0>>+(3*x3)*<<0,0,1,0>>+ ---snip ---*<<0,0,0,0>>,1]],
809: [dx1,dx2,dx3,dx4],
810: [(1)*<<0,0,0,2>>,(1)*<<0,0,1,0>>,(1)*<<0,0,0,1>>,(1)*<<0,0,0,0>>]]
811:
812: [2674] mt_gkz.dp_op_to_coef_vec([x1*<<1,0>>+x1*x2*<<0,1>>,x1+1],[<<1,0>>,<<0,1>>]);
813: // x1+1 is the denominator
814: [ (x1)/(x1+1) (x1*x2)/(x1+1) ]
815: [2675] mt_gkz.tk_base_is_equal([1,2],[1,2]);
816: 1
817: [2676] mt_gkz.tk_base_is_equal([1,2],[1,x,y]);
818: 0
819: [2677] mt_gkz.mdiff(sin(x),x,1);
820: cos(x)
821: [2678] mt_gkz.mdiff(sin(x),x,2); //2nd derivative
822: -sin(x)
823: [3164] mt_gkz.ord_xi(V=[x1,x2,x3],II=1);
824: // matrix to define graded lexicographic order so that V[II] is the smallest.
825: [ 1 1 1 ]
826: [ 0 -1 0 ]
827: [ -1 0 0 ]
828: [3166] load("mt_gkz/check-by-hgpoly.rr");
829: [3187] check_0123(); // check the pfaffian for the A below by hg-polynomial.
830: A=[[1,1,1,1],[0,1,2,3]]
831: Ap=[[1,1,1,1],[0,0,0,0]]
832: --- snip ---
833: Bfunctions= --- snip ---
834: 0 (vector) is expected:
835: [[ 0 0 0 ],[ 0 0 0 ]]
836: [3188] mt_gkz.get_check_fvec();
837: // get the basis of cocycles used in terms of differential operators.
838: [1,(dx4)/(b1),(dx4^2)/(b1^2-b1)]
839: [3189] mt_gkz.clear_bf();
840: 0
841: [3190] mt_gkz.get_bf_step_up(A=[[1,1,1,1],[0,1,2,3]]);
842: // b-functions and step-up operators.
843: // Option b_ideal=1 or shift=... may be used for non-normal case.
844: [[ -s_2^3+(9*s_1-3)*s_2^2+ ---snip---
845: -s_2^3+(3*s_1+1)*s_2^2-3*s_1*s_2 s_2^3-3*s_2^2+2*s_2 ],
846: [ x3^3*dx4^2+ ---snip---
847: 3*x3^2*x4*dx4^2+ --- snip---]]
1.2 takayama 848: [3191] mt_gkz.mytoric_ideal(0 | use_4ti2=1);
849: // 4ti2 is used to obtain a generator set of the toric ideal
850: // defined by the matrix A
851: [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
852: // A slower method is used to obtain a generator set of the toric ideal
853: // defined by the matrix A. 4ti2 is not needed. Default.
1.5 takayama 854: [3193] mt_gkz.cbase_by_euler(A=[[1,1,1,1],[0,1,3,4]]);
855: // Cohomology basis of the GKZ system defined by A for generic beta.
856: // Basis is given by a set of Euler operators.
857: // It is an implementation of the algorithm in http://dx.doi.org/10.1016/j.aim.2016.10.021
858: // beta is set by random numbers. Option: no_prob=1
859:
1.1 takayama 860: @end example
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916:
917: @comment ここから追加版
918:
919: @node Cohomology intersection numbers,,, Top
920: @chapter Cohomology intersection numbers
921:
922: @menu
923: * mt_gkz.kronecker_prd::
924: * mt_gkz.secondary_eq::
925: * mt_gkz.generate_maple_file_IC::
926: * mt_gkz.generate_maple_file_MR::
1.2 takayama 927: * mt_gkz.principal_normalizing_constant::
1.1 takayama 928: @end menu
929:
930:
931:
932:
933:
934:
935: @node Secondary equation,,, Cohomology intersection numbers
936: @section Secondary equation
937:
938: @comment **********************************************************
939: @comment --- 関数 pfaff_eq
940: @node mt_gkz.kronecker_prd,,, Secondary equation
941: @subsection @code{mt_gkz.kronecker_prd}
942: @comment --- 索引用キーワード
943: @findex mt_gkz.kronecker_prd
944:
945: @table @t
946: @item mt_gkz.kronecker_prd(@var{A},@var{B})
947: :: It returns the Kronecker product of @var{A} and @var{B}.
948: @end table
949:
950: @comment --- 引数の簡単な説明 ---
951: @table @var
952: @item return
1.2 takayama 953: a matrix which is equal to the Kronecker product of @var{A} and @var{B} (@uref{https://en.wikipedia.org/wiki/Kronecker_product}).
1.1 takayama 954: @item A,B
955: list
956: @end table
957:
958:
959: @comment --- @example〜@end example は実行例の表示 ---
960:
961: @example
962: [2644] A=[[a,b],[c,d]];
963: [[a,b],[c,d]]
964: [2645] B=[[e,f],[g,h]];
965: [[e,f],[g,h]]
1.2 takayama 966: [2646] mt_gkz.kronecker_prd(A,B);
1.1 takayama 967: [ e*a f*a e*b f*b ]
968: [ g*a h*a g*b h*b ]
969: [ e*c f*c e*d f*d ]
970: [ g*c h*c g*d h*d ]
971: @end example
972:
973:
974:
975:
976:
977:
978:
979:
980: @node mt_gkz.secondary_eq,,, Secondary equation
981: @subsection @code{mt_gkz.secondary_eq}
982: @comment --- 索引用キーワード
983: @findex mt_gkz.secondary_eq
984:
985: @table @t
986: @item mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
987: :: It returns the secondary equation with respect to cocycles defined by Rvec.
988: @end table
989:
990:
991: @table @var
992: @item return
993: a list of coefficients of the Pfaffian system corresponding to the secondary equation (cf. equation (8) of [MT2020]).
994: @item A,Beta,Ap,Rvec,DirX
995: see @code{pfaff_eq}
996: @end table
997:
998:
999:
1000: @comment --- ここで関数の詳しい説明 ---
1001: @comment --- @itemize〜@end itemize は箇条書き ---
1002: @comment --- @bullet は黒点付き ---
1003: @itemize @bullet
1004: @item
1005: The secondary equation is originally a Pfaffian system for an unkwon @math{r} by @math{r} matrix @math{I} with @math{r=}length(Rvec). We set @math{Y=(I_{11},I_{12},...,I_{1r},I_{21},I_{22},...)^T}. Then, the secondary equation can be seen as a Pfaffian system @math{{dY\over dx_i}=A_iY} with DirX=@math{\{dx_i\}_i}. The function mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX}) outputs a list obtained by aligning the matrices @math{A_i}.
1006: @item
1007: Let @math{F:=(\omega_i)_i} be a column vector whose entries are given by the cohomology classes specified by entries of Rvec. Then, @code{pfaff_eq} computes the Pfaffian matrices @math{P_i} so that @math{{dF\over dx_i}=P_iF}. If @math{Q_i} denotes the matrix obtained by replacing Beta by -Beta, we have @math{A_i=}@code{mt_gkz.kronecker_prd}(E,@math{P_i})+@code{mt_gkz.kronecker_prd}(@math{Q_i},E) where E is the identity matrix of size length(Rvec).
1008: @item Options xrule, shift, b_ideal,cg.
1009: Same as @code{pfaff_eq}.
1010: @end itemize
1011:
1012: @comment --- @example〜@end example は実行例の表示 ---
1013: Example:
1014: @example
1015: [2647] Beta=[b1,b2,b3]$
1016: [2648] DirX=[dx1,dx4]$
1017: [2649] Rvec=[[1,0,0,0],[0,0,1,0]]$
1018: [2650] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1019: [2651] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1020: [2652] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1021: [2653] P=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
1.1 takayama 1022: --snip--
1023: [2654] length(P);
1024: 2
1025: [2655] P[0];
1026: [[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),(b2*x4)/(x1^2*x4-x1),
1027: (-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
1028: ((b2-4/3)*x1^2*x4^2+(-b1-b2+8/3)*x1*x4+b1-4/3)/(x1^3*x4^2-2*x1^2*x4+x1),0,
1029: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
1030: ((-b2-2/3)*x1^2*x4^2+(b1+b2+4/3)*x1*x4-b1-2/3)/(x1^3*x4^2-2*x1^2*x4+x1),
1031: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]
1032: <--- Paffian matrix in x1 direction.
1033: [2656] P[1];
1034: [[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
1035: ((b2-1/3)*x1^2*x4^2+(-b1-b2+2/3)*x1*x4+b1-1/3)/(x1^2*x4^3-2*x1*x4^2+x4),0,
1036: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
1037: ((-b2+1/3)*x1^2*x4^2+(b1+b2-2/3)*x1*x4-b1+1/3)/(x1^2*x4^3-2*x1*x4^2+x4),
1038: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]]
1039: <--- Paffian matrix in x4 direction.
1040: @end example
1041:
1042: @comment --- 参照(リンク)を書く ---
1043: @table @t
1044: @item Refer to
1045: @ref{mt_gkz.pfaff_eq}
1046: @end table
1047: @comment おわり.
1048:
1049:
1050:
1051:
1052: @node mt_gkz.generate_maple_file_IC,,, Secondary equation
1053: @subsection @code{mt_gkz.generate_maple_file_IC}
1054: @comment --- 索引用キーワード
1055: @findex mt_gkz.generate_maple_file_IC
1056:
1057: @table @t
1058: @item mt_gkz.generate_maple_file_IC(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1059: :: It returns the maple input for a solver of a Pfaffian system IntegrableConnections[RationalSolutions].
1060: @end table
1061:
1062: @comment --- 引数の簡単な説明 ---
1063: @table @var
1064: @item return
1065: a maple input file for the function IntegrableConnections[RationalSolutions] (cf. [BCEW]) for the Pfaffian system mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX}).
1066: @item A,Beta,Ap,Rvec,DirX
1067: see @code{pfaff_eq}.
1068: @end table
1069:
1070: @comment --- ここで関数の詳しい説明 ---
1071: @comment --- @itemize〜@end itemize は箇条書き ---
1072: @comment --- @bullet は黒点付き ---
1073: @itemize @bullet
1074: @item
1075: A maple package IntegrableConnections is available in [BCEW]. In order to use IntegrableConnections, you need to add the global path to the file IntegrableConnections.m to libname on maple. See [BCEW].
1076: @item
1077: If Beta contains unkwon variables, they are regarded as generic parameters. For example, if Beta=[b1,b2,1/5,1/7,b5,...], parameters are [b1,b2,b5,...].
1078: @item Options xrule, shift, b_ideal,cg.
1079: Same as @code{pfaff_eq}.
1080: @item Option filename.
1081: You can specify the file name by specifying the option variable filename. If you do not specify it, @code{generate_maple_file_IC} generates a file "auto-generated-IC.ml".
1082: @end itemize
1083:
1084: @comment --- @example〜@end example は実行例の表示 ---
1085: Example:
1086: @example
1087: [2681] Beta=[b1,b2,1/3]$
1088: [2682] DirX=[dx1,dx4]$
1089: [2683] Rvec=[[1,0,0,0],[0,0,1,0]]$
1090: [2684] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1091: [2685] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1092: [2687] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1093: [2688] mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$
1.1 takayama 1094:
1095:
1096: //A file named Test.ml is automatically generated as follows:
1097:
1098:
1099:
1100: with(OreModules);
1101: with(IntegrableConnections);
1102: with(linalg);
1103: C:=[Matrix([[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),
1104: (b2*x4)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
1105: ((b2-4/3)*x1^2*x4^2+(-b1-b2+8/3)*x1*x4+b1-4/3)/(x1^3*x4^2-2*x1^2*x4+x1),0,
1106: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
1107: ((-b2-2/3)*x1^2*x4^2+(b1+b2+4/3)*x1*x4-b1-2/3)/(x1^3*x4^2-2*x1^2*x4+x1),
1108: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]),
1109: Matrix([[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
1110: ((b2-1/3)*x1^2*x4^2+(-b1-b2+2/3)*x1*x4+b1-1/3)/(x1^2*x4^3-2*x1*x4^2+x4),0,
1111: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
1112: ((-b2+1/3)*x1^2*x4^2+(b1+b2-2/3)*x1*x4-b1+1/3)/(x1^2*x4^3-2*x1*x4^2+x4),
1113: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]])];
1114: RatSols:=RationalSolutions(C,[x1,x4],['param',[b1,b2]]);
1115:
1116:
1117: /*
1118: If you run the output file on maple, you obtain a rational solution of
1119: the secondary equation.
1120: */
1121:
1122: [b2*(3*b1-1)/(b1*x1^2)]
1123: RatSols:=[3*b2/x1 ]
1124: [3*b2/x1 ]
1125: [3*b2-1 ]
1126:
1127: /*
1128: Note that the 4 entries of this vector correspond to entries of a 2 by 2 matrix.
1129: They are aligned as (1,1), (1,2), (2,1) (2,2) from the top.
1130: */
1131: @end example
1132:
1133: @*
1134:
1135: @comment --- 参照(リンク)を書く ---
1136: @table @t
1137: @item Refer to
1138: @ref{mt_gkz.pfaff_eq}
1139: @end table
1140: @comment おわり.
1141:
1142:
1143:
1144:
1145:
1146:
1147:
1148:
1149: @node mt_gkz.generate_maple_file_MR,,, Secondary equation
1150: @subsection @code{mt_gkz.generate_maple_file_MR}
1151: @comment --- 索引用キーワード
1152: @findex mt_gkz.generate_maple_file_MR
1153:
1154: @table @t
1155: @item mt_gkz.generate_maple_file_MR(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX},@var{D1},@var{D2})
1156: :: It returns the maple input for a solver of a Pfaffian system MorphismsRat[OreMorphisms].
1157: @end table
1158:
1159: @comment --- 引数の簡単な説明 ---
1160: @table @var
1161: @item return
1162: a maple input file for the function MorphismsRat[OreMorphisms] (cf. [CQ]) for the Pfaffian system obtained by @code{secondary_eq}. If you run the output file on maple, you obtain a rational solution of the secondary equation.
1163: @item A,Beta,Ap,Rvec,DirX
1164: see @code{pfaff_eq}.
1165: @item D1,D2
1166: Positive integers. D1 (resp. D2) is the upper bound of the degree of the numerator (resp. denominator) of the solution.
1167: @end table
1168:
1169: @comment --- ここで関数の詳しい説明 ---
1170: @comment --- @itemize〜@end itemize は箇条書き ---
1171: @comment --- @bullet は黒点付き ---
1172: @itemize @bullet
1173: @item
1174: We use the same notation as the explanation of @code{generate_maple_file_IC}. Let @math{D} denote the ring of linear differential operators with coeffiecients in the field of rational functions. We consider @math{D}-modules @math{R:=D^{1\times l}/\sum_{dx_i\in DirX}D^{1\times l}(\partial_i E-P_i)} and @math{S:=D^{1\times l}/\sum_{dx_i\in DirX}D^{1\times l}(\partial_i E+Q_i^T)} where @math{l=}length(Rvec). Then, computing a rational solution of the secondary equation is equivalent to computing a @math{D}-morphism from @math{R} to @math{S} represented by rational function matrix (cf. pp12-13 of [CQ08]).
1175: @item
1176: A maple package OreMorphisms is available in [CQ]. In order to use OreMorphisms, you need to add the global path to the file OreMorphisms.m to libname on maple.
1177: @item Options xrule, shift, b_ideal,cg.
1178: Same as @code{pfaff_eq}.
1179: @item Option filename.
1180: You can specify the file name as in @code{generate_maple_file_IC}.
1181: @item
1182: The difference between @code{generate_maple_file_IC} and @code{generate_maple_file_MR} is the appearence of auxilliary variables D1 and D2. If you can guess the degree of the numerator and the denominator of the solution of the secondary equation, MorphismsRat[OreMorphisms] can be faster than RationalSolutions[IntegrableConnections].
1183: @end itemize
1184:
1185: @comment --- @example〜@end example は実行例の表示 ---
1186: Example:
1187: @example
1188: [2668] Beta=[b1,b2,1/3]$
1189: [2669] DirX=[dx1,dx4]$
1190: [2670] Rvec=[[1,0,0,0],[0,0,1,0]]$
1191: [2671] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1192: [2672] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1193: [2673] Xvar=[x1,x4]$
1194: [2674] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1195: [2675] mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$
1.1 takayama 1196:
1197:
1198: //A file "auto-generated-MR.ml" is automatically generated as follows:
1199:
1200:
1201: with(OreModules);
1202: with(OreMorphisms);
1203: with(linalg);
1204: Alg:=DefineOreAlgebra(diff=[dx1,x1],diff=[dx4,x4],polynom=[x1,x4],comm=[b1,b2]);
1205: P:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
1206: -Matrix([[((b1+b2-4/3)*x1*x4-b1+4/3)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1)],
1207: [(-b1)/(x1*x4-1),(b1*x4)/(x1*x4-1)],[(b2*x1)/(x1*x4-1),(-b2)/(x1*x4-1)],
1208: [(-b1*x1)/(x1*x4^2-x4),(1/3*x1*x4+b1-1/3)/(x1*x4^2-x4)]]);
1209: Q:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
1210: +Matrix([[((-b1-b2-2/3)*x1*x4+b1+2/3)/(x1^2*x4-x1),(b1)/(x1*x4-1)],
1211: [(b2*x4)/(x1^2*x4-x1),(-b1*x4)/(x1*x4-1)],[(-b2*x1)/(x1*x4-1),(b1*x1)/(x1*x4^2-x4)],
1212: [(b2)/(x1*x4-1),(-1/3*x1*x4-b1+1/3)/(x1*x4^2-x4)]]);
1213: RatSols:=MorphismsRat(P,Q,Alg,0,2,2);
1214:
1215: /*
1216: If you run the output file on maple, you obtain a vector RatSols.
1217: RatSols[1] is the rational solution of the secondary equation:
1218: */
1219:
1220: RatSols[1]:=[(1/3)*@math{n_{2_{1_{3_1}}}}*(3*b1-1)/(b1*x1^2*@math{d_{6_1}}) @math{n_{2_{1_{3_1}}}}/(x1*@math{d_{6_1}})]
1221: [@math{n_{2_{1_{3_1}}}}/(x1*@math{d_{6_1}}) (1/3)*@math{n_{2_{1_{3_1}}}}*(3*b2-1)/(b2*@math{d_{6_1}})]
1222:
1223: /*
1224: Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are arbitrary constants. We can take @math{n_{2_{1_{3_1}}}=3*b2} and @math{d_{6_1}=1} to obtain the rational solution of the secondary equation which is identical to the one obtained from @code{generate_maple_file_IC}.
1225: */
1226: @end example
1227:
1228: @*
1229:
1230: @comment --- 参照(リンク)を書く ---
1231: @table @t
1232: @item Refer to
1233: @ref{mt_gkz.pfaff_eq}, @ref{mt_gkz.generate_maple_file_IC}.
1234: @end table
1235: @comment おわり.
1236:
1237:
1238:
1239:
1240:
1241:
1242:
1243:
1244:
1245:
1246:
1247:
1248: @node Normalizing constant,,, Cohomology intersection numbers
1249: @section Normalizing the cohomology intersection matrix
1250:
1251:
1252: @node mt_gkz.principal_normalizing_constant,,, Normalizing constant
1253: @subsection @code{mt_gkz.principal_normalizing_constant}
1254: @comment --- 索引用キーワード
1255: @findex mt_gkz.principal_normalizing_constant
1256:
1257: @table @t
1258: @item mt_gkz.principal_normalizing_constant(@var{A},@var{T},@var{Beta},@var{K})
1259: :: It returns the normalizing constant of the cohomology intersection matrix in terms of a regular triangulation T.
1260: @end table
1261:
1262: @comment --- 引数の簡単な説明 ---
1263: @table @var
1264: @item return
1.3 takayama 1265: a rational function which is the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[{dt\over t}],[{dt\over t}]\rangle_{ch}} in terms of the regular triangulation T. Here, @math{n} is the number of integration variables and @math{dt\over t} is the volume form @math{{dt_1\over t_1}\wedge\cdots\wedge{dt_n\over t_n}} of the complex @math{n}-torus.
1.1 takayama 1266: @item A,Beta
1267: see @code{pfaff_eq}.
1268: @item T
1269: a regular triangulation of A.
1270: @item K
1271: The number of polynomial factors in the integrand. see [MT2020].
1272: @end table
1273:
1274: @comment --- ここで関数の詳しい説明 ---
1275: @comment --- @itemize〜@end itemize は箇条書き ---
1276: @comment --- @bullet は黒点付き ---
1277: @itemize @bullet
1278: @item
1.3 takayama 1279: This function is useful when the basis of the cohomology group @math{\{\omega_i\}_{i=1}^r} is given so that @math{\omega_1=[{dt\over t}]}.
1.1 takayama 1280: @item
1.3 takayama 1281: One can find a regular triangulation by using a function @code{mt_gkz.regular_triangulation}.
1282: @item
1283: @code{mt_gkz.leading_terms} can be used more generally.
1.1 takayama 1284: @end itemize
1285:
1286: @comment --- @example〜@end example は実行例の表示 ---
1287: Example:
1288: @example
1289: [2676] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1290: [2677] Beta=[b1,b2,b3]$
1291: [2678] K=2$
1292: [2679] T=[[1,2,3],[2,3,4]]$
1.2 takayama 1293: [2680] mt_gkz.principal_normalizing_constant(A,T,Beta,K);
1.1 takayama 1294: (-b1-b2)/(b3*b1+b3*b2-b3^2)
1295: @end example
1296:
1297: @comment --- 参照(リンク)を書く ---
1298: @table @t
1299: @item Refer to
1.3 takayama 1300: @ref{mt_gkz.leading_terms}.
1.1 takayama 1301: @end table
1302: @comment おわり.
1303:
1304:
1305:
1306:
1307:
1308:
1309:
1310:
1.3 takayama 1311: @node mt_gkz.leading_terms,,, Normalizing constant
1312: @subsection @code{mt_gkz.leading_terms}
1.1 takayama 1313: @comment --- 索引用キーワード
1314: @findex mt_gkz.leading_terms
1315:
1316: @table @t
1.3 takayama 1317: @item mt_gkz.leading_terms(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N})
1.1 takayama 1318: :: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.
1319: @end table
1320:
1321: @comment --- 引数の簡単な説明 ---
1322: @table @var
1323: @item return
1.3 takayama 1324: a list [[C1,DEG1],[C2,DEG2],...]. Each CI is a rational function depending on Beta times a monomial @math{x^m} in x-variables. DEGI is the W-degree of @math{x^m}. The cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}t^{q_1^{\prime\prime}}{dt\over t}],[h^{-q_2^\prime}t^{q_2^{\prime\prime}}{dt\over t}]\rangle_{ch}} has a Laurent expansion of the form C1+C2+....
1.1 takayama 1325: @item A,Beta
1326: see @code{pfaff_eq}.
1327: @item W
1328: a positive and integral weight vector.
1329: @item Q1,Q2
1330: @math{Q1=(q_1^\prime,q_1^{\prime\prime})^T}, @math{Q2=(q_2^\prime,q_2^{\prime\prime})^T} are integer vectors. The lengths of @math{q_1^\prime} and @math{q_2^\prime} are both equal to @math{K}.
1331: @item K
1332: The number of polynomial factors in the integrand. see [MT2020].
1333: @item N
1334: A positive integer.
1335: @end table
1336:
1337: @comment --- ここで関数の詳しい説明 ---
1338: @comment --- @itemize〜@end itemize は箇条書き ---
1339: @comment --- @bullet は黒点付き ---
1340: @itemize @bullet
1341: @item
1342: For a monomial @math{x^m=x_1^{m_1}\cdots x_n^{m_n}} and a weight vector @math{W=(w_1,\dots,w_n)}, the W-degree of @math{x^m} is given by the dot product @math{m\cdot W=m_1w_1+\cdots +m_nw_n}.
1343: @item
1.3 takayama 1344: The W-leading terms of the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}t^{q_1^{\prime\prime}}{dt\over t}],[h^{-q_2^\prime}t^{q_2^{\prime\prime}}{dt\over t}]\rangle_{ch}} can be computed by means of Theorem 2.6 of [GM2020]. See also Theorem 3.4.2 of [SST2000].
1.1 takayama 1345: @item
1346: If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". In this case, the output may be wrong and you should retake a suitable W. To be more precise, W should be chosen from an open cone of the Groebner fan.
1347: @item Option xrule.
1348: Same as @code{pfaff_eq}.
1349: @end itemize
1350:
1351: @comment --- @example〜@end example は実行例の表示 ---
1352: Example:
1353: @example
1354: [2922] Beta=[b1,b2,1/3];
1355: [b1,b2,1/3]
1356: [2923] Q=[[1,0,0],[0,1,0]];
1357: [[1,0,0],[0,1,0]]
1358: [2924] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];
1359: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
1360: [2925] W=[1,0,0,0];
1361: [1,0,0,0]
1362: [2926] K=2;
1363: 2
1364: [2927] N=2;
1365: 2
1.3 takayama 1366: [2928] NC=mt_gkz.leading_terms(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$
1.1 takayama 1367: --snip--
1368: [2929] NC;
1369: [[(-3)/(x1),-5],[0,-4],[0,-3]]
1370:
1371:
1372: /*
1373: This output means that the W-leading term of the (1,2) entry of the cohomology
1374: intersection matrix is @math{(-3)/(x1)\times (2\pi\sqrt{-1})}. In view of examples of @code{generate_maple_file_IC} or @code{generate_maple_file_MR}, we can conclude that the cohomology
1375: intersection matrix is given by
1376: */
1377:
1378: [-(3*b1-1)/(b1*x1^2) -3/x1 ]
1379: [-3/x1 -(3*b2-1)/b2]]
1380:
1381:
1382: //divided by 2@math{\pi\sqrt{-1}}.
1383: @end example
1384:
1385:
1386: @comment --- 参照(リンク)を書く ---
1387: @table @t
1388: @item Refer to
1.3 takayama 1389: @ref{mt_gkz.leading_terms}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}.
1.1 takayama 1390: @end table
1391: @comment おわり.
1392:
1393:
1394:
1395:
1396:
1397: @node mt_gkz.leading_term_rat,,, Normalizing constant
1398: @subsection @code{mt_gkz.leading_term_rat}
1399: @comment --- 索引用キーワード
1400: @findex mt_gkz.leading_term_rat
1401:
1402: @table @t
1403: @item mt_gkz.leading_term_rat(@var{P},@var{W},@var{V})
1404: :: It returns the W-leading term of a rational function P depending on variables V.
1405: @end table
1406:
1407: @comment --- 引数の簡単な説明 ---
1408: @table @var
1409: @item return
1410: It returns the W-leading term of a rational function P.
1411: @item P
1412: a rational function.
1413: @item W
1414: a weight vector.
1415: @item V
1416: a list of variables of P.
1417: @end table
1418:
1419: @comment --- ここで関数の詳しい説明 ---
1420: @comment --- @itemize〜@end itemize は箇条書き ---
1421: @comment --- @bullet は黒点付き ---
1422: @itemize @bullet
1423: @item
1.3 takayama 1424: This function is supposed to be combined with @code{leading_terms} to compute the leading term of a cohomology intersection number.
1.1 takayama 1425: @item
1426: If W is chose so that there are several initial terms, you will receive an error message "WARNING(leading_term_rat):The weight vector may not be generic."
1427: @end itemize
1428:
1429:
1430: @comment --- 参照(リンク)を書く ---
1431: @table @t
1432: @item Refer to
1.3 takayama 1433: @ref{mt_gkz.leading_terms}.
1.1 takayama 1434: @end table
1435: @comment おわり.
1436:
1437:
1438:
1439:
1440:
1441:
1442: @node Regular triangulations,,, Cohomology intersection numbers
1443: @section Regular triangulations
1444:
1445: @comment --- 個々の関数の説明 ---
1446: @comment --- section 名を正確に ---
1447: @node mt_gkz.toric_gen_initial,,, Regular triangulations
1448: @node mt_gkz.regular_triangulation,,, Regular triangulations
1449: @node mt_gkz.top_standard_pairs,,, Regular triangulations
1450: @subsection @code{mt_gkz.toric_gen_initial}, @code{mt_gkz.regular_triangulation}, @code{mt_gkz.top_standard_pairs}
1451: @comment --- 索引用キーワード
1452: @findex mt_gkz.toric_gen_initial
1453: @findex mt_gkz.regular_triangulation
1454: @findex mt_gkz.top_standard_pairs
1455:
1456: @table @t
1457: @item mt_gkz.toric_gen_initial(@var{A},@var{W})
1458: @item mt_gkz.regular_triangulation(@var{A},@var{W})
1459: @item mt_gkz.top_standard_pairs(@var{A},@var{W})
1460: :: utility functions for computing ring theoretic invariants: generic initial ideal for the toric ideal specified by the matrix A and a weight W, its associated regular triangulation, and its associated top-dimensional standard pairs.
1461: @end table
1462:
1463: @comment --- 引数の簡単な説明 ---
1464: @table @var
1465: @item return
1466: @code{toric_gen_initial} returns a list [L1,L2] of length 2. L1 is a list of generators of the W-initial ideal of the toric ideal @math{I_A} specified by A. L2 is a list of variables of @math{I_A}.
1467: @item return
1468: @code{regular_triangulation} returns a list of simplices of a regular triangulation @math{T_W} specified by the weight W.
1469: @item return
1470: @code{top_standard_pairs} returns a list of the form [[L1,S1],[L2,S2],...]. Each SI is a simplex of @math{T_W}. Each LI is a list of exponents.
1471: @item A
1472: a configuration matrix.
1473: @item W
1474: a positive weight vector.
1475: @end table
1476:
1477: @comment --- ここで関数の詳しい説明 ---
1478: @comment --- @itemize〜@end itemize は箇条書き ---
1479: @comment --- @bullet は黒点付き ---
1480: @itemize @bullet
1481: @item
1482: As for the definition of the standard pair, see Chapter 3 of [SST00].
1483: @item
1484: We set n=length(A) and set BS1:=@math{\{ 1,2,...,n\}\setminus S1}. Then, each L1[I] is an exponent @math{\bf k} of a top-dimensional standard pair @math{(\partial^{\bf k}_{BS1},S1)}. Here, @math{\bf k} is a list of length n-length(S1) and @math{\partial_{BS1}=(\partial_J)_{J\in BS1}}.
1485: @item
1.3 takayama 1486: If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". See also @code{leading_terms}.
1.1 takayama 1487: @item
1.3 takayama 1488: These functions are utilized in @code{leading_terms}.
1.1 takayama 1489: @end itemize
1490:
1491: @comment --- @example〜@end example は実行例の表示 ---
1492: Example: An example of a non-unimodular triangulation and non-trivial standard pairs.
1493: @example
1494: [3256] A=[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]];
1495: [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
1496: [3257] W=[2,0,1,2,2];
1497: [2,0,1,2,2]
1.3 takayama 1498: [3258] mt_gkz.toric_gen_initial(A,W);
1.1 takayama 1499: --snip--
1500: [[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
1.3 takayama 1501: [3259] mt_gkz.regular_triangulation(A,W);
1.1 takayama 1502: --snip--
1503: [[2,4,5],[2,3,5],[1,2,3]]
1.3 takayama 1504: [3260] mt_gkz.top_standard_pairs(A,W);
1.1 takayama 1505: --snip--
1506: [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]
1507:
1508: /*
1509: This means that the regular triangulation of the configuration matrix A is
1510: given by @math{T=\{\{2,4,5\},\{2,3,5\},\{1,2,3\}\}}. The normalized volumes of these simplices
1511: are 2,1 and 1. Moreover, the top-dimensional standard pairs are
1512: @math{(1,\{2,4,5\}),(\partial_3,\{2,4,5\})}, @math{(1,\{2,3,5\})},@math{(1,\{1,2,3\})}.
1513: */
1514: @end example
1515:
1516:
1517:
1518: @comment --- 参照(リンク)を書く ---
1519: @table @t
1520: @item Refer to
1.3 takayama 1521: @ref{mt_gkz.leading_terms}.
1.1 takayama 1522: @end table
1523: @comment おわり.
1524:
1525:
1526:
1527:
1528:
1529:
1530:
1531: @comment --- おまじない ---
1532: @node Index,,, Top
1533: @unnumbered Index
1534: @printindex fn
1535: @printindex cp
1536: @iftex
1537: @vfill @eject
1538: @end iftex
1539: @summarycontents
1540: @contents
1541: @bye
1542: @comment --- おまじない終り ---
1543:
1544: @comment *********************************************************
1545: @comment ********* template
1546: @comment **********************************************************
1547: @comment --- 個々の関数の説明 ---
1548: @comment --- section 名を正確に ---
1549: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
1550: @subsection @code{mt_gkz.pfaff_eq}
1551: @comment --- 索引用キーワード
1552: @findex mt_gkz.pfaff_eq
1553:
1554: @table @t
1555: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1556: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
1557: @end table
1558:
1559: @comment --- 引数の簡単な説明 ---
1560: @table @var
1561: @item return
1562: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
1563: @item A
1564: the matrix A of the GKZ system.
1565: @item Beta
1566: ...
1567: @end table
1568:
1569: @comment --- ここで関数の詳しい説明 ---
1570: @comment --- @itemize〜@end itemize は箇条書き ---
1571: @comment --- @bullet は黒点付き ---
1572: @itemize @bullet
1573: @item
1574: The independent variables are @code{x1,x2,x3,...} (@math{x_1, x_2, x_3, \ldots}).
1575: @end itemize
1576:
1577: @comment --- @example〜@end example は実行例の表示 ---
1578: Example: Gauss hypergeometric system, see [GM2020] example ??.
1579: @example
1580: [1883] import("mt_gkz.rr");
1581: @end example
1582:
1583: @comment --- 参照(リンク)を書く ---
1584: @table @t
1585: @item Refer to
1586: @ref{mt_gkz.pfaff_eq}
1587: @end table
1588: @comment おわり.
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