GKZ hypergeometric system

Pfaffian system (Pfaff equation), contiguity relations, cohomology intersection

Version 1.0

December 15, 2022

Copyright © Risa/Asir committers 2004–2020. All rights reserved.

1 About this document

This document explains Risa/Asir functions for GKZ hypergeometric system (A-hypergeometric system).
Loading the package:

import("mt_gkz.rr");

References cited in this document.

• [MT2020] Saiei-Jaeyeong Matsubara-Heo, Nobuki Takayama, Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals, Lecture Notes in Computer Science 12097 (2020), 73–84. Errata is posted on http://arxiv.org/abs/???. E-attachments can be obtainable at https://www.math.kobe-u.ac.jp/OpenXM/Math/intersection2
• [GM2020] Yoshiaki Goto, Saiei-Jaeyeong Matsubara-Heo, Homology and cohomology intersection numbers of GKZ systems, arXiv:2006.07848
• [SST1999] M.Saito, B.Sturmfels, N.Takayama, Hypergeometric polynomials and integer programming, Compositio Mathematica, 155 (1999), 185–204
• [SST2000] M.Saito, B.Sturmfels, N.Takayama, Groebner Deformations of Hypergeometric Differential Equations. Springer, 2000.
• [HNT] T.Hibi, K.Nishiyama, N.Takayama, Pfaffian systems of A-Hypergeometric equations I: bases of twisted cohomology groups. Advances in Mathematics, 306 (2017), 303–327. https://dx.doi.org/10.1016/j.aim.2016.10.021

References for maple packages IntegrableConnections and OreMorphisms.

• [BCEW] M.Barkatou, T.Cluzeau, C.El Bacha, J.-A.Weil, IntegrableConnections – a maple package for computing closed form solutions of integrable connections (2012). https://www.unilim.fr/pages perso/thomas.cluzeau/Packages/IntegrableConnections/PDS.html
• [CQ] T.Cluzeau and A.Quadrat, OreMorphisms: A homological algebraic package for factoring, reducing and decomposing linear functional systems (2009). https://who.rocq.inria.fr/Alban.Quadrat/OreMorphisms/index.html
• [CQ08] T.Cluzeau, A.Quadrat, Factoring and decomposing a class of linear functional systems, Linear Algebra and its Applications (LAA), 428(1): 324-381, 2008.

2 Pfaff equation

2.1 Pfaff equation for given cocycles

2.1.1 mt_gkz.pfaff_eq

mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,DirX)

:: It returns the Pfaff equation for the GKZ system defined by A and Beta with respect to cocycles defined by Rvec.

return

a list of coefficients of the Pfaff equation with respect to the direction DirX

A

the matrix A of the GKZ system.

Beta

the parameter vector of the GKZ system.

Ap

See [MT2020].

Rvec

It is used to specify a basis of cocycles as explained below. See also [MT2020].

DirX

a list of dxi’s.

• The independent variables are x1, x2, x3, ...
• When Rvec=[v_1, v_2, ..., v_r] where r is the rank of the GKZ system, the set of the cocycles standing for Av_1, Av_2, ..., Av_r (see [MT2020]) is supposed to be the basis to construct the Pfaffian system. The exponents (q_\ell, q) of the integral representation \int \prod h_\ell^{-q_\ell} x^q {{dx} \over {x}} is shifted by Av_i=:A_{v_i} as (q_\ell,q)+A_{v_i}. Let a_1, a_2, ..., a_n be the column vectors of the matrix A and v be a column vector (x_1, x_2, ..., x_n)^T. Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n.
• When the columns of A are expressed as e_i \otimes \alpha_{i_j}, the columns of Ap is e_i \otimes 0 where $e_i$ is the i-th unit vector. See [MT2020] on the definition of Ap. Here are some examples. When A is

  [[1,1,0,0],
   [0,0,1,1],
   [0,1,0,1]]
  

  Ap is

  [[1,1,0,0],
   [0,0,1,1],
   [0,0,0,0]] <-- zero row
  

  When A is

  [[1,1,1,0,0,0],
   [0,0,0,1,1,1],
   [0,1,0,0,1,0],
   [0,0,1,0,0,1]
  ]
  

  Ap is

  [[1,1,1,0,0,0],
   [0,0,0,1,1,1],
   [0,0,0,0,0,0], <-- zero row
   [0,0,0,0,0,0]  <-- zero row
  ]
  

  See also page 223 of [SST2000].

• Option xrule. When the option xrule is given, the x variables specified by this option are specialized to numbers.
• Option shift. When the matrix A is not normal (the associated toric ideal is not normal), a proper shift vector must be given to obtain an element of the b-ideal. Or, use the option b_ideal below. See [SST1999] on the theory.
• Option b_ideal. When the matrix A is not normal, the option b_ideal=1 obtains b-ideals and the first element of each b-ideal is used as the b-function. The option shift is ignored.
• Option cg. A constant matrix given by this option is used for the Gauge transformation of the Pfaffian system. In other words, the basis of cocycles specified by Rvec is transformed by the constant matrix given by this option.
• By mt_gkz.use_hilbert_driven(Rank), the rank of the GKZ system is assumed to be Rank. It makes the computation of Groebner basis by yang.rr faster. This option is disabled by mt_gkz.use_hilbert_driven(0);

Example: Gauss hypergeometric system, see [GM2020] example ??.

[1883] import("mt_gkz.rr");
[2657] PP=mt_gkz.pfaff_eq(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
             Beta=[-g1,-g2,-c],
             Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
             Rvec = [[1,0,0,0],[0,0,1,0]], 
             DirX=[dx4,dx3] | xrule=[[x1,1],[x2,1]], 
             cg=matrix_list_to_matrix([[1,0],[-1,1]]))$

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]
-- snip --
[2658] PP[0];
[ (g2*x3-g2)/(x4-x3) (g2*x3)/(x4-x3) ]
[ ((-g2*x3-c+g2)*x4+(c-g1)*x3+g1)/(x4^2-x3*x4) 
           ((-g2*x3-c)*x4+(c-g1)*x3)/(x4^2-x3*x4) ]
[2659] PP[1];
[ (-g2*x4+g2)/(x4-x3) (-g2*x4)/(x4-x3) ]
[ ((g2*x3+c-g2-1)*x4+(-c+g1+1)*x3-g1)/(x3*x4-x3^2) 
     ((g2*x3+c-g2-1)*x4+(-c+g1+g2+1)*x3)/(x3*x4-x3^2) ]

Example: The role of shift. When the toric ideal is not normal, a proper shift vector must be given with the option shift to find an element of the b-ideal.

[1882] load("mt_gkz.rr");
[1883] A=[[1,1,1,1],[0,1,3,4]];
  [[1,1,1,1],[0,1,3,4]]
[1884] Ap=[[1,1,1,1],[0,0,0,0]];
  [[1,1,1,1],[0,0,0,0]]
[1885] Rvec=[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
  [[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
[2674] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
    | xrule=[[x1,1],[x2,2],[x3,4]] )$
dx remains
stopped in step_up at line 342 in file "./mt_gkz/saito-b.rr"
342    if (type(dn(Ans)) > 1) error("dx remains");
(debug) quit
// Since the toric ideal for A is not normal, it stops with the error.
[2675]  P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4] 
   | shift=[1,0],xrule=[[x1,1],[x2,2],[x3,4]])$
// It works.

Refer to

mt_gkz.ff1 mt_gkz.ff2 mt_gkz.ff2, mt_gkz.ff1, mt_gkz.ff mt_gkz.rvec_to_fvec

2.1.2 mt_gkz.ff2, mt_gkz.ff1, mt_gkz.ff

mt_gkz.ff(Rvec0,A,Ap,Beta)

mt_gkz.ff1(Rvec0,A,Beta,Ap)

mt_gkz.ff2(Rvec0,A,Beta,Ap,BF,C)

:: ff returns a differential operator whose action to 1 gives the cocycle defined by Rvec0

return

ff returns a differential operator whose action to 1 of M_A(\beta) gives the cocycle defined by Rvec0.

return

ff1 returns a composite of step-down operators for the positive part of Rvec0

return

ff2 returns a composite of step-up operators for the positive part of Rvec0

Rvec0

An element of Rvec explained in mt_gkz.pfaff_eq.

BF

the list of b-functions to all directions.

C

the list of the step up operators for all a_1, a_2, ..., a_n.

Other arguments are same with those of pfaff_eq.

• The function ff generates the list of b-functions and the list of step up operators and store them in the cache variable. They can be obtained by calling as S=mt_gkz.get_bf_step_up() where S[0] is the list of b-functions and S[1] is the list of step up operators. Step up operators are obtained by the algorithm given in [SST1999].
• Option nf. When nf=1, the output operator is reduced to the normal form with respect to the Groebner basis of the GKZ system of the graded reverse lexicographic order.
• Option shift. See mt_gkz.pfaff_eq.
• Internal info: The function mt_gkz.bb gives the constant so that the step up and step down operators (contiguity operators) give contiguity relations for the integral representation in [MT2020]. Note that mt_gkz.ff1 and mt_gkz.ff2 give contiguity relations which are constant multiple of those for hypergeometric polynomials.
• Internal info: mt_gkz.step_up generates step up operators of [SST1999] from b-functions by utilizing mt_gkz.bf2euler and mt_gkz.toric.

Example: Step up operators compatible with the integral representation in [MT2020]. The function hgpoly_res defined in check-by-hgpoly.rr returns a multiple of the hypergeometric polynomial which agrees with the residue times a power of 2\pi \sqrt{-1} of the integral representation. See [SST1999].

[1883] import("mt_gkz.rr")$
[3175] load("mt_gkz/check-by-hgpoly.rr")$
[3176]  A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[3177]  B=newvect(3,[5,4,7])$ Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
[3179]  Beta=[b1,b2,b3]$  R=[0,0,-1,0]$
[3180]  F2=hgpoly_res(A,B,2);  // HG polynomial. 2 is the number of e_i's
                               // =the number of non-zero rows of Ap
  10*x1^2*x2^3*x4^4+20*x1*x2^4*x3*x4^3+6*x2^5*x3^2*x4^2
[3182]  mt_gkz.ff(R,A,Ap,Beta); // the operator standing for R
  (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
[3184] S=mt_gkz.get_bf_step_up(A); // b-function and non-reduced step up op's
  [[ 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 ],
  [ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1 
    x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2 
    x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3 
    x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]
[3185] Fvec=mt_gkz.ff2(R,A,Beta,Ap,S[0],S[1]);
  (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
[3188] Fvec = base_replace(Fvec,assoc(Beta,vtol(B)));
  1/3*x3*x4*dx4+1/3*x3^2*dx3+1/3*x1*x4*dx2+1/3*x1*x3*dx1+1/3*x3
[3189] R32d = odiff_act(Fvec,F2,[x1,x2,x3,x4]); // Act Fvec to the hg-poly
  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
[3190] red(R32d/hgpoly_res(A,B+newvect(3,[0,1,0]),2)); 
   // R32d agrees with the HG polynomial with Beta=[5,4,7]+[0,1,0].
1

Refer to

mt_gkz.pfaff_eq

2.1.3 mt_gkz.rvec_to_fvec

mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)

:: It returns a set of differential operators standing for Rvec.

return

It returns a set of differential operators of which action to 1 \in M_A(\beta) give cocycles specified by Rvec.

A, Ap, Beta

Same with mt_gkz.pfaff_eq

• Internal info: this function builds the set of operators by calling mt_gkz.ff2, mt_gkz.ff1, mt_gkz.ff.

Example: The following two expressions are congruent because 2a_1-a_2-a_3+a_4=a_1 for this A.

[1883] import("mt_gkz.rr");
[3191] mt_gkz.rvec_to_fvec([[2,-1,-1,1],[0,0,1,0]],
 [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
 [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
[(x2*x3*x4^2*dx1^2*dx4^3+((x1*x3*x4^2+x2*x3^2*x4)*dx1^2*dx3
 +(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
 +(x1*x4^2+3*x2*x3*x4)*dx1^2)*dx4^2+(x1*x3^2*x4*dx1^2*dx3^2
 +((x1^2*x3*x4+x1*x2*x3^2)*dx1^3+(3*x1*x3*x4+x2*x3^2)*dx1^2)*dx3
 +x1*x2^2*x4*dx1^2*dx2^2+((x1^2*x2*x4+x1*x2^2*x3)*dx1^3
 +(3*x1*x2*x4+x2^2*x3)*dx1^2)*dx2+x1^2*x2*x3*dx1^4
 +(x1^2*x4+3*x1*x2*x3)*dx1^3+(x1*x4+x2*x3)*dx1^2)*dx4)
 /(b3*b2*b1^3+(b3*b2^2+(-b3^2-2*b3)*b2)*b1^2+(-b3*b2^2+(b3^2+b3)*b2)*b1),
 (dx3)/(b2)]
[3192] mt_gkz.rvec_to_fvec([[1,0,0,0],[0,0,1,0]],
   [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
   [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
[(dx1)/(b1),(dx3)/(b2)]

Refer to

mt_gkz.pfaff_eq

2.1.4 mt_gkz.fvec_to_conn_mat

mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,DirX)

:: It returns the coefficient matrices of the basis Fvec or DirX[I]*Fvec in terms of the set of the standard basis.

return

It returns the coefficient matrices of the basis Fvec or DirX[I]*Fvec in terms of the set of the standard basis of the Groebner basis explained below.

A Beta

Same with mt_gkz.pfaff_eq.

DirX

When DirX is 1, this function returns the matrix which expresses Fvec in terms of the set of the standard monomials of the Groebner basis of the GKZ system in the ring of rational function coefficients with respect to the graded reverse lexicographic order. In other cases, it returns the coefficient matrices of DirX[I]’s*Fvec in terms of the set of the standard basis of the Groebner basis.

• It utilizes a Groebner basis computation by the package yang.rr and yang.reduction to obtain connection matrices.
• This function calls some utility functions mt_gkz.dmul(Op1,Op2,XvarList) (multiplication of Op1 and Op2 and mt_gkz.index_vars(x,Start,End | no_=1) which generates indexed variables without the underbar “_”.
• We note here some other utility functions in this section: mt_gkz.check_compatibility(P,Q,X,Y), which checkes if the sytem d/dX-P, d/dY-Q is compatible.

Example: The following example illustrates how mt_gkz.pfaff_eq obtains connection matrices.

[1883] import("mt_gkz.rr");
[3201] V=mt_gkz.index_vars(x,1,4 | no_=1);
  [x1,x2,x3,x4]
[3202] mt_gkz.dmul(dx1,x1^2,V);
  x1^2*dx1+2*x1
[3204] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
   Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
   Beta= [b1,b2,b3]$
   Rvec = [[1,0,0,0],[0,0,1,0]]$
   Fvec = mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)$
    /* Express cocyles Rvec 
       by elements Fvec in the Weyl algebra by contiguity relations. */
   Cg = matrix_list_to_matrix([[1,0],[1,-1]])$
[3208] NN=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,1); 
  // Express Fvec by the standard monomials Std=NN[1].
 1 ooo 2 .ooo
  [[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
  [ (-x4)/(b1*x2) (1)/(x3) ],[dx4,1]]
[3209] Std=NN[1]; 
  [dx4,1]
[3173] NN=NN[0];
  [ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
  [ (-x4)/(b2*x3) (1)/(x3) ]
[3174] NN1=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,dx1)[0];
  // Express dx1*Fvec by the standard monomials Std.
 1 ooo 2 .ooo
  [ ((2*b1+b2-b3-1)*x1*x4^2+(-b1+b3+1)*x2*x3*x4)/(b1*x1^3*x4-b1*x1^2*x2*x3)
    ((b1^2+(-2*b3-1)*b1-b3*b2+b3^2+b3)*x1*x4
     +(-b1^2+(2*b3+1)*b1-b3^2-b3)*x2*x3)/(b1*x1^3*x4-b1*x1^2*x2*x3) ]
  [(b1 (-b1*x1*x4^2-b2*x2*x3*x4)/(b2*x1^2*x3*x4-b2*x1*x2*x3^2) 
   (b1*x1*x4+(-b1+b3)*x2*x3)/(x1^2*x3*x4-x1*x2*x3^2) ]
[3188] P1=map(red,Cg*NN1*matrix_inverse(NN)*matrix_inverse(Cg));
  [ ((-b2*x3+(b1+b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3)/(x1^2*x4-x1*x2*x3) 
     (b2*x3*x4)/(x1^2*x4-x1*x2*x3) ]
  [ ((-b2*x3+(b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3+b1*x1*x2)/(x1^2*x4-x1*x2*x3) 
    ((b2*x3+b1*x1)*x4)/(x1^2*x4-x1*x2*x3) ]

[3191] mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,[dx1]|cg=Cg)[0]-P1;
 [ 0 0 ]
 [ 0 0 ]  // P1 agrees with the output of mt_gkz.pfaff_eq.

Refer to

mt_gkz.pfaff_eq

2.1.5 mt_gkz.contiguity, mt_gkz.contiguity_by_fvec

mt_gkz.contiguity(A,Beta,Ap,Rvec1,Rvec2)

:: It returns the coefficient matrix P that satisfies Rvec1 = P Rvec2.

mt_gkz.contiguity_by_fvec(A,Beta,Ap,Fvec1,Fvec2)

:: It returns the coefficient matrix P that satisfies Fvec1 = P Fvec2.

return

The coefficient matrix P that satisfies Rvec1 = P Rvec2 or Fvec1=P Fvec2

A Beta Ap Rvec1 Rvec2

Same with mt_gkz.pfaff_eq.

• It returns the contiguity relation between Rvec1 and Rvec2

Example:

[1883] import("mt_gkz.rr");
[3200] PP=mt_gkz.contiguity(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
             Beta=[-g1,-g2,-c],
             Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
             Rvec1 = [[1,0,0,0],[0,0,1,0]],   
             Rvec2 = [[0,0,1,0],[1,0,0,0]]);   

[3366] Fvec411=mt_gkz.rvec_to_fvec(Rvec411=[[1,1,0]], 
                    A=[[1,1,1],[1,0,1],[0,1,1]],
                    Ap=[[1,1,1],[0,0,0],[0,0,0]],
                    Beta=[eps,-eps*del,-eps*del])$
Fvec411d=[mt_gkz.dmul(dx1,Fvec411[0],[x1,x2,x3])]; 
 [(dx1^2*dx2)/(eps^2-eps)]
[3367] mt_gkz.contiguity_by_fvec(A,Beta,Ap,Fvec411d,Fvec411);  
 1 .ooo
[ ((del+1)*eps-1)/(x1) ]

Refer to

mt_gkz.pfaff_eq mt_gkz.fvec_to_conn_mat

3 b function

3.1 b function and facet polynomial

3.1.1 mt_gkz.bf

mt_gkz.bf(A,Facet_poly,II0)

:: It returns the b-function with respect to the direction II0.

return

It returns the b-function introduced Saito with respect to the direction II0 in case of A is normal or an element of b-ideal when a proper shift vector is given in case of A is not normal.

A

the matrix A of the GKZ system.

Facet_poly

The set of facet polynomials of the convex hull of A.

II0

Direction expressed as 0, 1, 2, ... (not 1, 2, 3, ...) to obtain the b function.

• See [SST1999] on the b-function introduced Saito and b-ideal.
• The facet polynomial must be primitive.

Example:

[1883] import("mt_gkz.rr");

[3193] A;
  [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
[3194] Fpoly=mt_gkz.facet_poly(A);
  [[s_3,s_1,s_2-s_3+s_1,s_2],[[0,0,1],[1,0,0],[1,1,-1],[0,1,0]]]
[3196] mt_gkz.bf(A,Fpoly,0);
  s_1*s_2-s_1*s_3+s_1^2
[3197] mt_gkz.bf(A,Fpoly,1);
  s_1*s_3

Refer to

mt_gkz.ff2, mt_gkz.ff1, mt_gkz.ff mt_gkz.facet_poly

3.1.2 mt_gkz.facet_poly

mt_gkz.facet_poly(A)

:: It returns the set of facet polynomials and their normal vectors of the cone defined by A.

return

It returns the set of facet polynomials and their normal vectors of the cone generated by the column vectors of the matrix A.

A

the matrix A of the GKZ system.

• The facet polynomial f is primitive. In other words, all f(a_i) is integer and min f(a_i)=1 for a_i’s not being on f=0. where a_i is the i-th column vector of the matrix A. It can be checked by mt_gkz.is_primitive(At,Facets) where At is the transpose of A and Facets is the second return value of this function.
• This function utilizes the system polymake https://polymake.org on our server.

Example:

[1883] import("mt_gkz.rr");
[1884] mt_gkz.facet_poly([[1,1,1,1],[0,1,2,3]]);
  oohg_native=0, oohg_curl=1
  [[s_2,-s_2+3*s_1],[[0,1],[3,-1]]]

Refer to

mt_gkz.bf

3.1.3 mt_gkz.get_bf_step_up

mt_gkz.get_bf_step_up(A)

:: It returns the set of b-functions and corresponding step-up operators for A.

return

The first entry is a list of b-function for all direction. The second entry is a list of step-up operators (quasi inverse of step-down operators dx_i’s).

A

the matrix A of the GKZ system.

• It is an implementation of the algorithm to obtain contiguity relations in [SST1999] M.Saito, B.Sturmfels, N.Takayama, Hypergeometric polynomials and integer programming, Compositio Mathematica, 155 (1999), 185–204.

Example:

[1883] import("mt_gkz.rr");;
[3176]  A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];;
[3177] S=mt_gkz.get_bf_step_up(A); 
  [[ 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 ],
  [ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1 
    x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2 
    x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3 
    x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]

Refer to

mt_gkz.bf

4 Utilities

4.1 Some utility functions

4.1.1 mt_gkz.gkz_b

mt_gkz.gkz_b(A,Beta,Simplex)

:: It returns the so called Horn system (B expression of GKZ) standing for the GKZ system for A, Beta.

return

the Horn system. It is expressed by the Euler operator, e.g., <<3>> stands for (x1*dx1)^3 in D.

A

the matrix A of the GKZ system.

Beta

the vector beta of the GKZ system.

Simplex

the indices of the variables to put 1. The index starts from 1. It should be a simplex obtained by a regular triangulation.

• When the option partial=1 is given, the Horn system is expressed by partial differential operators, e.g., <<1,2>> stands for dx1*dx2^2.
• When B is the Gale transform matrix of A, in other words, AB=0 where the column vectors span Ker A, the associated Horn system is obtained from the series expansion associated to the matrix B and the simplex. The ’b’ of the function name gkz_b comes from this fact.

Example: Pfaffian equation for Appell function F1.

import("mt_gkz.rr");
import("yang.rr");
A=[[1,1,1,0,0,0],
   [0,0,0,1,1,1],
   [0,1,0,0,1,0],
   [0,0,1,0,0,1]];
Id=mt_gkz.gkz_b(A,[b1,b2,b3,b4],[1,2,3,6] | partial=1);
yang.define_ring(["partial",Xvars=[x4,x5]]);
// Id=map(mt_mm.remove_redundancy,Id,Xvars);
yang.verbose();
G=yang.buchberger(Id);
Std=yang.stdmon(G);
Pf=yang.pfaffian(map(dp_ptod,Std,poly_dvar(Xvars)),G);
// print_xdvi_form(poly_factor(Pf[0]);
end$

Outputs are, e.g.,

[3104] Id[2];
(x4*x5^2-x4*x5)*<<1,1>>+(x5^3-x5^2)*<<0,2>>+(-b3*x4*x5)*<<1,0>>
+((-b2-b3+1)*x5^2+(b2-b4-1)*x5)*<<0,1>>+(b3*b2*x5)*<<0,0>>
[3105] G[1];
[(-x5+x4)*<<1,1>>+(b3)*<<1,0>>+(-b1-b2+b3+b4)*<<0,1>>,-x5+x4]
[3106] Std;
[dx4,dx5,1]

Pf[0] by print_tex_form(poly_factor(Pf[0])) is
\pmatrix{ { ( ( ( {b}_{1}+ 2 {b}_{2}- 2 {b}_{3}- {b}_{4}- 1) {x}_{4}- {b}_{2}+ {b}_{3}+ {b}_{4}+ 1) {x}_{5}+ ( - {b}_{1}- 2 {b}_{2}+ {b}_{3}+ {b}_{4}+ 1) {x}_{4}^{ 2} + ( {b}_{2}- {b}_{4}- 1) {x}_{4})}\over{ ( {x}_{4}) ( {x}_{4}- 1) ( {x}_{5}- {x}_{4})}& { ( {b}_{1}+ {b}_{2}- {b}_{3}- {b}_{4}) ( {x}_{5}) ( {x}_{5}- 1)}\over { ( {x}_{4}) ( {x}_{4}- 1) ( {x}_{5}- {x}_{4})}& { ( - 1) ( {b}_{2}) ( {b}_{1}+ {b}_{2}- {b}_{3}- {b}_{4})}\over { ( {x}_{4}) ( {x}_{4}- 1)} \cr { ( {b}_{3})} \over{ ( {x}_{5}- {x}_{4})}& { ( - 1) ( {b}_{1}+ {b}_{2}- {b}_{3}- {b}_{4})} \over{ ( {x}_{5}- {x}_{4})}& 0 \cr 1& 0& 0 \cr } . dx4-Pf[0], dx5-Pf[1] are Pfaffian equations obtained by the Buchberger algorithm.

Refer to

mt_gkz.regular_triangulation mt_gkz.toric_gen_initial, mt_gkz.regular_triangulation, mt_gkz.top_standard_pairs

4.1.2 mt_gkz.pf_ratsol_ml

mt_gkz.pf_ratsol_ml(Pf,V,Param)

:: It returns a Maple code to find rational solutions of d/dV[0]-Pf[0], d/dV[1]-Pf[1], ...

return

The name of the auto generated maple program. The default is “auto-generated-pf-ratsol.ml”.

Pf

A list of Pfaffian matrices.

V

A list of independent variables.

Param

A list of parameter variables.

• The option ntsetting=1 assumes the Maple libraries OreModules and IntegrableConnections are under $HOME/Math/Maple_lib

Example: rational solutions of Pfaffian equation of two variables. The Pfaffian equation is for

[2529] F=sm1.mul(x*dx+y*dy+a,x*dx+1,[x,y,a])-x*(x*dx+1);
x^2*dx^2+(y*x*dy-x^2+(a+2)*x)*dx+y*dy-x+a
[2530] G=sm1.mul(x*dx+y*dy+a,y*dy+1,[x,y,a]);
(y*x*dy+x)*dx+y^2*dy^2+(a+2)*y*dy+a

It has a rational solution 1/(x*y). Input the following codes.

import("mt_gkz.rr");
Px=[[(x-a-1)/(x),0,(x-a+1)/(x^2)],[(-1)/(y),(-1)/(x),(-1)/(y*x)],[1,0,0]];
Py=[[(-1)/(y),(-1)/(x),(-1)/(y*x)],[0,(-a-1)/(y),(-a+1)/(y^2)],[0,1,0]];
mt_gkz.pf_ratsol_ml([ Px, Py ], [x,y],[a] );

Then, outputs are, e.g.,

[3033] mt_gkz.pf_ratsol_ml([ Px, Py ], [x,y],[a] | ntsetting=1);
auto-generated-pf-ratsol.ml


#Load this program by read(`auto-generated-pf-ratsol.ml`);
libname:="/home/nobuki/Math/Maple_lib", libname;
with(OreModules);
with(IntegrableConnections);
with(linalg);
C:=[Matrix([[(x-a-1)/(x),0,(x-a+1)/(x^2)],[(-1)/(y),(-1)/(x),(-1)/(y*x)],[1,0,0]]),
 Matrix([[(-1)/(y),(-1)/(x),(-1)/(y*x)],[0,(-a-1)/(y),(-a+1)/(y^2)],[0,1,0]])];
RatSols:=RationalSolutions(C,[x,y],['param',[a]]);

Refer to

mt_gkz.generate_maple_file_IC

4.1.3 mt_gkz.cbase_by_euler

mt_gkz.cbase_by_euler(A)

:: It returns a standard basis for the GKZ system defined by A.

return

The list of a standard basis for the GKZ system defined by A.

A

the matrix A of the GKZ system.

option w

w is a weight vector (given by a list) for differential operators.

option no_prob

If it is set to 1, a probabilistic algorithm of setting b’s to random numbers is not used.

• The returned list of differential operators (monomials) is a standard basis of the GKZ system and it also gives a cohomology basis by applying them to the integration kernel of the GKZ hypergeometric integral.
• It is an implementation of the algorithm in [HNT] https://dx.doi.org/10.1016/j.aim.2016.10.021 by Hibi-Nishiyama-Takayama.
• The weight vector is used to obtain an initial ideal of the toric ideal for A and is also used to obtain a standard basis of a system of algebraic equations by [HNT].

Example: basis for A=[[1,1,1,1],[0,1,2,3]].

import("mt_gkz.rr");
A=[[1,1,1,1],[0,1,2,3]]$
Std1=mt_gkz.cbase_by_euler(A);
Std2=mt_gkz.cbase_by_euler(A | w=[0,0,1,1]);
end$

Outputs are, e.g.,

[3079] A=[[1,1,1,1],[0,1,2,3]]$

[3080] Std1=mt_gkz.cbase_by_euler(A);
Gt=[-dx1*dx3+dx2^2,dx1*dx4-dx2*dx3,-dx2*dx4+dx3^2]
T2     =[dx2^2,dx2*dx3,dx3^2]
T2(new)=[dx2^2-dx2,dx2*dx3,dx3^2-dx3]
E=[dx4+dx3+dx2+dx1-b_1,3*dx4+2*dx3+dx2-b_2]
We use a probablistic method to determine the base.
Note: dx1, dx2, ..., are the Euler operators
[dx3,dx4,1]

[3081] Std2=mt_gkz.cbase_by_euler(A | w=[0,0,1,1]);
T2     =[dx2*dx3,dx1*dx3,dx1^2*dx4,dx3^2]
T2(new)=[dx2*dx3,dx1*dx3,(dx1^2-dx1)*dx4,dx3^2-dx3]
E=[dx4+dx3+dx2+dx1-b_1,3*dx4+2*dx3+dx2-b_2]
We use a probablistic method to determine the base.
Note: dx1, dx2, ..., are the Euler operators
[dx2^2,dx2,1]
[3082] 

Refer to

@ref{mt_gkz.mytoric_by_euler}

For other functions, we have not yet written manual entries. We only show examples on these. As for details, please see the source code.

[1883] import("mt_gkz.rr");
[2667] mt_gkz.dvar([x1,x2]);  // it generates variables starting with d
  [dx1,dx2]
[2669] mt_gkz.p_true_nf_rat((1/3)*x^3-1,[x^2-1],[x],0);
  [x-3,3]     // p_true_nf does not accept rational number coefficients
[2670] A=[[1,1,1,1],[0,1,3,4]];
  [[1,1,1,1],[0,1,3,4]]
[2671] mt_gkz.reduce_by_toric(dx3^4,A);
  dx1*dx4^3   // reduction by toric ideal defined by A
[2672] nk_toric.toric_ideal(A);
  [-x1*x4+x2*x3,-x2*x4^2+x3^3,x2^2*x4-x1*x3^2,-x1^2*x3+x2^3]
[2673] mt_gkz.yang_gkz_buch(A,[b1,b2]);  // Groebner basis of GKZ system by yang.rr
 1 o 2 ..o 3 ..oooooooo 4 o 6 ooo 9 o
[[[(x2)*<<0,1,0,0>>+(3*x3)*<<0,0,1,0>>+ ---snip ---*<<0,0,0,0>>,1]],
[dx1,dx2,dx3,dx4],
[(1)*<<0,0,0,2>>,(1)*<<0,0,1,0>>,(1)*<<0,0,0,1>>,(1)*<<0,0,0,0>>]]

[2674] mt_gkz.dp_op_to_coef_vec([x1*<<1,0>>+x1*x2*<<0,1>>,x1+1],[<<1,0>>,<<0,1>>]);
  // x1+1 is the denominator
  [ (x1)/(x1+1) (x1*x2)/(x1+1) ]
[2675] mt_gkz.tk_base_is_equal([1,2],[1,2]);
 1
[2676] mt_gkz.tk_base_is_equal([1,2],[1,x,y]);
 0
[2677] mt_gkz.mdiff(sin(x),x,1);
 cos(x)
[2678] mt_gkz.mdiff(sin(x),x,2); //2nd derivative
 -sin(x)
[3164] mt_gkz.ord_xi(V=[x1,x2,x3],II=1); 
// matrix to define graded lexicographic order so that V[II] is the smallest.
[ 1 1 1 ]
[ 0 -1 0 ]
[ -1 0 0 ]
[3166] load("mt_gkz/check-by-hgpoly.rr");
[3187] check_0123(); // check the pfaffian for the A below by hg-polynomial.
  A=[[1,1,1,1],[0,1,2,3]]
  Ap=[[1,1,1,1],[0,0,0,0]]
  --- snip ---
  Bfunctions= --- snip ---
  0 (vector) is expected:
  [[ 0 0 0 ],[ 0 0 0 ]]
[3188] mt_gkz.get_check_fvec();  
 // get the basis of cocycles used in terms of differential operators.
  [1,(dx4)/(b1),(dx4^2)/(b1^2-b1)]
[3189] mt_gkz.clear_bf();
0
[3190] mt_gkz.get_bf_step_up(A=[[1,1,1,1],[0,1,2,3]]);
// b-functions and step-up operators.
// Option b_ideal=1 or shift=... may be used for non-normal case.
  [[ -s_2^3+(9*s_1-3)*s_2^2+ ---snip---
     -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 ],
  [ x3^3*dx4^2+ ---snip---
    3*x3^2*x4*dx4^2+ --- snip---]]
[3191] mt_gkz.mytoric_ideal(0 | use_4ti2=1);
// 4ti2 is used to obtain a generator set of the toric ideal 
// defined by the matrix A
[3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
// A slower method is used to obtain a generator set of the toric ideal 
// defined by the matrix A.  4ti2 is not needed. Default.
[3193] mt_gkz.cbase_by_euler(A=[[1,1,1,1],[0,1,3,4]]);
// Cohomology basis of the GKZ system defined by A for generic beta. 
// Basis is given by a set of Euler operators. 
// It is an implementation of the algorithm in https://dx.doi.org/10.1016/j.aim.2016.10.021
// beta is set by random numbers. Option: no_prob=1

5 Cohomology intersection numbers

5.1 Secondary equation

5.1.1 mt_gkz.kronecker_prd

mt_gkz.kronecker_prd(A,B)

:: It returns the Kronecker product of A and B.

return

a matrix which is equal to the Kronecker product of A and B (https://en.wikipedia.org/wiki/Kronecker_product).

A,B

list

[2644]  A=[[a,b],[c,d]];
[[a,b],[c,d]]
[2645] B=[[e,f],[g,h]];
[[e,f],[g,h]]
[2646] mt_gkz.kronecker_prd(A,B);
[ e*a f*a e*b f*b ]
[ g*a h*a g*b h*b ]
[ e*c f*c e*d f*d ]
[ g*c h*c g*d h*d ]

5.1.2 mt_gkz.secondary_eq

mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX)

:: It returns the secondary equation with respect to cocycles defined by Rvec.

return

a list of coefficients of the Pfaffian system corresponding to the secondary equation (cf. equation (8) of [MT2020]).

A,Beta,Ap,Rvec,DirX

see pfaff_eq

• The secondary equation is originally a Pfaffian system for an unkwon r by r matrix I with r=length(Rvec). We set Y=(I_{11},I_{12},...,I_{1r},I_{21},I_{22},...)^T. Then, the secondary equation can be seen as a Pfaffian system {dY\over dx_i}=A_iY with DirX=\{dx_i\}_i. The function mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX) outputs a list obtained by aligning the matrices A_i.
• Let F:=(\omega_i)_i be a column vector whose entries are given by the cohomology classes specified by entries of Rvec. Then, pfaff_eq computes the Pfaffian matrices P_i so that {dF\over dx_i}=P_iF. If Q_i denotes the matrix obtained by replacing Beta by -Beta, we have A_i=mt_gkz.kronecker_prd(E,P_i)+mt_gkz.kronecker_prd(Q_i,E) where E is the identity matrix of size length(Rvec).
• Options xrule, shift, b_ideal,cg. Same as pfaff_eq.

Example:

[2647] Beta=[b1,b2,b3]$
[2648] DirX=[dx1,dx4]$
[2649] Rvec=[[1,0,0,0],[0,0,1,0]]$
[2650] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[2651] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
[2652] Xrule=[[x2,1],[x3,1]]$
[2653] P=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
--snip--
[2654] length(P);
2
[2655] P[0];
[[(-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),
(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
((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,
(-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
((-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),
(b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]
<--- Paffian matrix in x1 direction.
[2656] P[1];
[[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
((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,
(-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
((-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),
(b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]]
<--- Paffian matrix in x4 direction.

Refer to

mt_gkz.pfaff_eq

5.1.3 mt_gkz.generate_maple_file_IC

mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX)

:: It returns the maple input for a solver of a Pfaffian system IntegrableConnections[RationalSolutions].

return

a maple input file for the function IntegrableConnections[RationalSolutions] (cf. [BCEW]) for the Pfaffian system mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX).

A,Beta,Ap,Rvec,DirX

see pfaff_eq.

• 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].
• 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,...].
• Options xrule, shift, b_ideal,cg. Same as pfaff_eq.
• Option filename. You can specify the file name by specifying the option variable filename. If you do not specify it, generate_maple_file_IC generates a file "auto-generated-IC.ml".

Example:

[2681] Beta=[b1,b2,1/3]$
[2682] DirX=[dx1,dx4]$
[2683] Rvec=[[1,0,0,0],[0,0,1,0]]$
[2684] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[2685] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
[2687] Xrule=[[x2,1],[x3,1]]$
[2688] mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$


//A file named Test.ml is automatically generated as follows:



with(OreModules);
with(IntegrableConnections);
with(linalg);
C:=[Matrix([[(-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),(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
((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,
(-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
((-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),
(b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]),
Matrix([[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
((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,
(-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
((-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),
(b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]])];
RatSols:=RationalSolutions(C,[x1,x4],['param',[b1,b2]]);


/*
If you run the output file on maple, you obtain a rational solution of
the secondary equation.
*/

         [b2*(3*b1-1)/(b1*x1^2)]
RatSols:=[3*b2/x1              ]
         [3*b2/x1              ]
         [3*b2-1               ]

/*
Note that the 4 entries of this vector correspond to entries of a 2 by 2 matrix.
They are aligned as (1,1), (1,2), (2,1) (2,2) from the top.
*/

Refer to

mt_gkz.pfaff_eq mt_gkz.gkz_b

5.1.4 mt_gkz.generate_maple_file_MR

mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,D1,D2)

:: It returns the maple input for a solver of a Pfaffian system MorphismsRat[OreMorphisms].

return

a maple input file for the function MorphismsRat[OreMorphisms] (cf. [CQ]) for the Pfaffian system obtained by secondary_eq. If you run the output file on maple, you obtain a rational solution of the secondary equation.

A,Beta,Ap,Rvec,DirX

see pfaff_eq.

D1,D2

Positive integers. D1 (resp. D2) is the upper bound of the degree of the numerator (resp. denominator) of the solution.

• We use the same notation as the explanation of generate_maple_file_IC. Let D denote the ring of linear differential operators with coeffiecients in the field of rational functions. We consider D-modules R:=D^{1\times l}/\sum_{dx_i\in DirX}D^{1\times l}(\partial_i E-P_i) and S:=D^{1\times l}/\sum_{dx_i\in DirX}D^{1\times l}(\partial_i E+Q_i^T) where l=length(Rvec). Then, computing a rational solution of the secondary equation is equivalent to computing a D-morphism from R to S represented by rational function matrix (cf. pp12-13 of [CQ08]).
• 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.
• Options xrule, shift, b_ideal,cg. Same as pfaff_eq.
• Option filename. You can specify the file name as in generate_maple_file_IC.
• The difference between generate_maple_file_IC and 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].

Example:

[2668] Beta=[b1,b2,1/3]$
[2669] DirX=[dx1,dx4]$
[2670] Rvec=[[1,0,0,0],[0,0,1,0]]$
[2671] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[2672] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
[2673] Xvar=[x1,x4]$
[2674] Xrule=[[x2,1],[x3,1]]$
[2675] mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$


//A file "auto-generated-MR.ml" is automatically generated as follows:


with(OreModules);
with(OreMorphisms);
with(linalg);
Alg:=DefineOreAlgebra(diff=[dx1,x1],diff=[dx4,x4],polynom=[x1,x4],comm=[b1,b2]);
P:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
-Matrix([[((b1+b2-4/3)*x1*x4-b1+4/3)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1)],
[(-b1)/(x1*x4-1),(b1*x4)/(x1*x4-1)],[(b2*x1)/(x1*x4-1),(-b2)/(x1*x4-1)],
[(-b1*x1)/(x1*x4^2-x4),(1/3*x1*x4+b1-1/3)/(x1*x4^2-x4)]]);
Q:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
+Matrix([[((-b1-b2-2/3)*x1*x4+b1+2/3)/(x1^2*x4-x1),(b1)/(x1*x4-1)],
[(b2*x4)/(x1^2*x4-x1),(-b1*x4)/(x1*x4-1)],[(-b2*x1)/(x1*x4-1),(b1*x1)/(x1*x4^2-x4)],
[(b2)/(x1*x4-1),(-1/3*x1*x4-b1+1/3)/(x1*x4^2-x4)]]);
RatSols:=MorphismsRat(P,Q,Alg,0,2,2);

/*
If you run the output file on maple, you obtain a vector RatSols.
RatSols[1] is the rational solution of the secondary equation:
*/

RatSols[1]:=[(1/3)*n_{2_{1_{3_1}}}*(3*b1-1)/(b1*x1^2*d_{6_1})  n_{2_{1_{3_1}}}/(x1*d_{6_1})]
            [n_{2_{1_{3_1}}}/(x1*d_{6_1})       (1/3)*n_{2_{1_{3_1}}}*(3*b2-1)/(b2*d_{6_1})]

/*
Here, n_{2_{1_{3_1}}} and d_{6_1} are arbitrary constants. We can take n_{2_{1_{3_1}}}=3*b2 and d_{6_1}=1 to obtain the rational solution of the secondary equation which is identical to the one obtained from generate_maple_file_IC.
*/

Refer to

mt_gkz.pfaff_eq, mt_gkz.generate_maple_file_IC.

5.2 Normalizing the cohomology intersection matrix

5.2.1 mt_gkz.principal_normalizing_constant

mt_gkz.principal_normalizing_constant(A,T,Beta,K)

:: It returns the normalizing constant of the cohomology intersection matrix in terms of a regular triangulation T.

return

a rational function which is the cohomology intersection number {1\over (2\pi\sqrt{-1})^n} \langle[{dt\over t}],[{dt\over t}]\rangle_{ch} in terms of the regular triangulation T. Here, n is the number of integration variables and dt\over t is the volume form {dt_1\over t_1}\wedge\cdots\wedge{dt_n\over t_n} of the complex n-torus.

A,Beta

see pfaff_eq.

T

a regular triangulation of A.

K

The number of polynomial factors in the integrand. see [MT2020].

• This function is useful when the basis of the cohomology group \{\omega_i\}_{i=1}^r is given so that \omega_1=[{dt\over t}].
• One can find a regular triangulation by using a function mt_gkz.regular_triangulation.
• mt_gkz.leading_terms can be used more generally.

Example:

[2676] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[2677] Beta=[b1,b2,b3]$
[2678] K=2$
[2679] T=[[1,2,3],[2,3,4]]$
[2680] mt_gkz.principal_normalizing_constant(A,T,Beta,K);
(-b1-b2)/(b3*b1+b3*b2-b3^2)

Refer to

mt_gkz.leading_terms.

5.2.2 mt_gkz.leading_terms

mt_gkz.leading_terms(A,Beta,W,Q1,Q2,K,N)

:: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.

return

a list [[C1,DEG1],[C2,DEG2],...]. Each CI is a rational function depending on Beta times a monomial x^m in x-variables. DEGI is the W-degree of x^m. The cohomology intersection number {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+....

A,Beta

see pfaff_eq.

W

a positive and integral weight vector.

Q1,Q2

Q1=(q_1^\prime,q_1^{\prime\prime})^T, Q2=(q_2^\prime,q_2^{\prime\prime})^T are integer vectors. The lengths of q_1^\prime and q_2^\prime are both equal to K.

K

The number of polynomial factors in the integrand. see [MT2020].

N

A positive integer.

• For a monomial x^m=x_1^{m_1}\cdots x_n^{m_n} and a weight vector W=(w_1,\dots,w_n), the W-degree of x^m is given by the dot product m\cdot W=m_1w_1+\cdots +m_nw_n.
• The W-leading terms of the cohomology intersection number {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].
• 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.
• Option xrule. Same as pfaff_eq.

Example:

[2922] Beta=[b1,b2,1/3];
[b1,b2,1/3]
[2923] Q=[[1,0,0],[0,1,0]];
[[1,0,0],[0,1,0]]
[2924] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];
[[1,1,0,0],[0,0,1,1],[0,1,0,1]]
[2925] W=[1,0,0,0];
[1,0,0,0]
[2926] K=2;
2
[2927] N=2;
2
[2928] NC=mt_gkz.leading_terms(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$
--snip--
[2929] NC;
[[(-3)/(x1),-5],[0,-4],[0,-3]]


/*
This output means that the W-leading term of the (1,2) entry of the cohomology
intersection matrix is (-3)/(x1)\times (2\pi\sqrt{-1}). In view of examples of generate_maple_file_IC or generate_maple_file_MR, we can conclude that the cohomology
intersection matrix is given by
*/

[-(3*b1-1)/(b1*x1^2)  -3/x1        ]
[-3/x1                -(3*b2-1)/b2]]


//divided by  2\pi\sqrt{-1}.

Refer to

mt_gkz.leading_terms, mt_gkz.generate_maple_file_IC, mt_gkz.generate_maple_file_MR.

5.2.3 mt_gkz.leading_term_rat

mt_gkz.leading_term_rat(P,W,V)

:: It returns the W-leading term of a rational function P depending on variables V.

return

It returns the W-leading term of a rational function P.

P

a rational function.

W

a weight vector.

V

a list of variables of P.

• This function is supposed to be combined with leading_terms to compute the leading term of a cohomology intersection number.
• 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."

Refer to

mt_gkz.leading_terms.

5.3 Regular triangulations

5.3.1 mt_gkz.toric_gen_initial, mt_gkz.regular_triangulation, mt_gkz.top_standard_pairs

mt_gkz.toric_gen_initial(A,W)

mt_gkz.regular_triangulation(A,W)

mt_gkz.top_standard_pairs(A,W)

:: 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.

return

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 I_A specified by A. L2 is a list of variables of I_A.

return

regular_triangulation returns a list of simplices of a regular triangulation T_W specified by the weight W.

return

top_standard_pairs returns a list of the form [[L1,S1],[L2,S2],...]. Each SI is a simplex of T_W. Each LI is a list of exponents.

A

a configuration matrix.

W

a positive weight vector.

• As for the definition of the standard pair, see Chapter 3 of [SST00].
• We set n=length(A) and set BS1:=\{ 1,2,...,n\}\setminus S1. Then, each L1[I] is an exponent \bf k of a top-dimensional standard pair (\partial^{\bf k}_{BS1},S1). Here, \bf k is a list of length n-length(S1) and \partial_{BS1}=(\partial_J)_{J\in BS1}.
• 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 leading_terms.
• These functions are utilized in leading_terms.

Example: An example of a non-unimodular triangulation and non-trivial standard pairs.

[3256] A=[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]];
[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
[3257] W=[2,0,1,2,2];
[2,0,1,2,2]
[3258] mt_gkz.toric_gen_initial(A,W);
--snip--
[[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
[3259] mt_gkz.regular_triangulation(A,W);
--snip--
[[2,4,5],[2,3,5],[1,2,3]]
[3260] mt_gkz.top_standard_pairs(A,W);
--snip--
[[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]

/*
This means that the regular triangulation of the configuration matrix A is
given by T=\{\{2,4,5\},\{2,3,5\},\{1,2,3\}\}. The normalized volumes of these simplices
are 2,1 and 1. Moreover, the top-dimensional standard pairs are
(1,\{2,4,5\}),(\partial_3,\{2,4,5\}), (1,\{2,3,5\}),(1,\{1,2,3\}).
*/

Refer to

mt_gkz.leading_terms.

Index

Table of Contents

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