Macaulay matrix method for GKZ hypergeometric systems

Pfaffian system (Pfaff equation) and Restriction

Version 0.7

March 26, 2023

by S-J. Matsubara-Heo, N.Takayama

Copyright © Risa/Asir committers 2004–2023. All rights reserved.

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1 About this document  
2 Pfaff equation  
3 Invariant subvector space  
4 Moser reduction::  
5 Restriction::  
Index  

1 About this document

This document explains Risa/Asir functions for Macaulay matrix method for GKZ hypergeometric systems (A-hypergeometric systems).
Loading the package: 

import("mt_mm.rr");

References cited in this document.

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2 Pfaff equation

2.1 mt_mm.find_macaulay  
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2.1 mt_mm.find_macaulay

mt_mm.find_macaulay(Ideal,Std,Xvars)

:: It returns a Macaulay type matrix for the Ideal with respect to the standard basis Std.

return

Data for Macaulay matrix.

Ideal

Generators of an ideal. It should be a list of distributed polynomials.

Std

Standard basis of the ideal Ideal. It should be a list of distributed polynomials.

Xvars

Independent variables

Option

deg, which is the starting degree of the Macaulay matrix.

Option

p, which is the prime number for the probabilistic rank check.

Option

restriction_var, which constructs a Macaulay matrix to obtain Pfaffian equations for restriction automatically.

Option

restriction_cond, which is a non-automatic setting to construct a Macaulay matrix for restriction.

Example: Macaulay matrix of x_1 \partial_1 + 2 x_2 \partial_2 -b_1, \partial_1^2-\partial_2 with respect to the standard monomials 1, \partial_2. The PDE is the GKZ system for the 1 by 2 matrix A=[1,2]. 

/* We input the followings */
import("mt_mm.rr");
Ideal=[x1*dx1+2*x2*dx2-b1,dx1^2-dx2]$
Std=[<<0,0>>,<<0,1>>]$ Xvars=[x1,x2]$
Id = map(dp_ptod,Ideal,poly_dvar(Xvars));
MData=mt_mm.find_macaulay(Id,Std,Xvars);
P1=mt_mm.find_pfaffian(MData,Xvars,1 | use_orig=1);
P2=mt_mm.find_pfaffian(MData,Xvars,2 | use_orig=1);

[3404] 
--- snip ---
[3405] [[(b1)/(x1),(-2*x2)/(x1)],
        [(1/2*b1^2-1/2*b1)/(x1*x2),((-b1+1)*x2-1/2*x1^2)/(x1*x2)]]  // P1
[3406] [[0,1],
        [(-1/4*b1^2+1/4*b1)/(x2^2),((b1-3/2)*x2+1/4*x1^2)/(x2^2)]] // P2

Note that Macaulay matrix method might give a fake answer under some situations. For example, we input the following non-integrable system. In other words, the Ideal below is the trivial ideal. We also input a wrong standard basis Std. 

/* We input the followings */
Ideal=[x1*dx1+x2*dx2-b1,dx1^2-dx2]$
Std=[<<0,0>>,<<0,1>>]$ Xvars=[x1,x2]$
Id = map(dp_ptod,Ideal,poly_dvar(Xvars));
MData=mt_mm.find_macaulay(Id,Std,Xvars);
P1=mt_mm.find_pfaffian(MData,Xvars,1 | use_orig=1);
P2=mt_mm.find_pfaffian(MData,Xvars,2 | use_orig=1);
nd_weyl_gr(Ideal,[x1,x2,dx1,dx2],0,0);

[3240] import("mt_mm.rr");
--- snip ---
[3542] MData=mt_mm.find_macaulay(Id,Std,Xvars);

We use a probabilistic method for rank_check with P=65537
Warning: prules are ignored.
--- snip ---
NT_info=[rank,[row, col]( of m=transposed MData[0]),indep_columns(L[2])]
Rank=6
The matrix is full rank.
We use a probabilistic method to compute the rank of a matrix with entries of rational functions.
The output is a macaulay matrix of degree 1
--- snip ---
[rank=6,[row,col]=[6,6] (size of t(M1)),Indep_cols] is stored in the variable NT_info.
--- snip ---
[[[0,0,0,0,0,x1],[0,0,1,0,0,0],[0,0,0,x1,x2,0],[0,1,0,0,-1,0],
 [0,0,x1,x2,0,-b1+1],[1,0,0,-1,0,0]],
[[-b1,x2],[0,-1],[0,-b1+1],[0,0],[0,0],[0,0]],
[(1)*<<3,0>>,(1)*<<2,1>>,(1)*<<2,0>>,(1)*<<1,1>>,(1)*<<0,2>>,(1)*<<1,0>>],
[(1)*<<0,0>>,(1)*<<0,1>>]]

[3543] P1=mt_mm.find_pfaffian(MData,Xvars,1 | use_orig=1);
[[(b1)/(x1),(-x2)/(x1)],[(b1^2-b1)/(x1*x2),((-b1+1)*x2-x1^2)/(x1*x2)]]

// d/dx1 - P1
[3544] P2=mt_mm.find_pfaffian(MData,Xvars,2 | use_orig=1);
[[0,1],[(-b1^2+b1)/(x2^2),((2*b1-2)*x2+x1^2)/(x2^2)]]
// d/dx2 - P2
[2545] nd_weyl_gr(Ideal,[x1,x2,dx1,dx2],0,0);
[-1]  // But it is a trivial ideal!

Let us go back to our running example. Homogenity is reduced by mt_gkz.gkz_b. Id0 is an ODE. 

/* We input the followings. */
A=[[1,2]]$ Beta=[b1]$ Sigma=[2]$
Id0=mt_gkz.gkz_b(A,Beta,Sigma|partial=1)$
Xvars=[x2]$
Id=map(mt_mm.remove_redundancy,Id0,Xvars);
Std=[1,dx2]$ Std=map(dp_ptod,Std,[dx2])$
MData=mt_mm.find_macaulay(Id,Std,Xvars);
P1=mt_mm.find_pfaffian(MData,Xvars,1 | use_orig=1);



[3244] A=[[1,2]]$ Beta=[b1]$ Sigma=[2]$
Id0=mt_gkz.gkz_b(A,Beta,Sigma|partial=1)$
Xvars=[x2]$
Id=map(mt_mm.remove_redundancy,Id0,Xvars);
[3245] [3246] [3247] [3248] 
[3249] [(x1^2)*<<2>>+(1/2*x1^3)*<<1>>+(-1/2*b1*x1^2)*<<0>>]
      // This is the ODE. 

[3252] Std=[1,dx2]$ Std=map(dp_ptod,Std,[dx2])$
MData=mt_mm.find_macaulay(Id,Std,Xvars);

[3253] [3254] We use a probabilistic method for rank_check with P=65537
Warning: prules are ignored.
--- snip ---
NT_info=[rank,[row, col]( of m=transposed MData[0]),indep_columns(L[2])]
Rank=2
The matrix is full rank.
We use a probabilistic method to compute the rank of a matrix with entries of rational functions.
The output is a macaulay matrix of degree 1
To draw a picture of the rref of the Macaulay matrix, use Util/show_mat with tmp-mm-*-M-transposed*.bin
[Std,Xvars] is stored in tmp-mm-9823-Std-Xvars.ab
[ES,Xvars] is stored in tmp-mm-9823-Es-Xvars.ab
[rank=2,[row,col]=[2,2] (size of t(M1)),Indep_cols] is stored in the variable NT_info.
NT_info is stored in tmp-mm-9823-NT_info.ab
MData is stored in tmp-mm-9823-mdata.ab

[[[0,x1^2],[x1^2,1/2*x1^3]],
 [[-1/2*b1*x1^2,1/2*x1^3],[0,-1/2*b1*x1^2]],
 [(1)*<<3>>,(1)*<<2>>],
 [(1)*<<0>>,(1)*<<1>>]]
[3255] P1=mt_mm.find_pfaffian(MData,Xvars,1 | use_orig=1);
[[0,1],[1/2*b1,-1/2*x1]]
// d/d<<1>> - P1
[3256]

When the parameters and independent variables are numbers, we can call find_pfaffian_fast. 

/* We input the followings. */
A=[[1,2]]$ Beta=[1/2]$ Sigma=[2]$
Id0=mt_gkz.gkz_b(A,Beta,Sigma|partial=1)$
Xvars=[x2]$
Id=map(mt_mm.remove_redundancy,Id0,Xvars)$
Std=[1,dx2]$ Std=map(dp_ptod,Std,[dx2])$
MData=mt_mm.find_macaulay(Id,Std,Xvars);
P1=mt_mm.find_pfaffian_fast(MData,Xvars,1,mt_mm.get_indep_cols() | xrules=[[x1,1/3]]);

--- snip ---
[3300] Generate a data for linsolv.
Sol rank=2
cmd=time /home/nobuki/bin/linsolv   --vars tmp-mm-9823-linsolv-vars.txt  <tmp-mm-9823-linsolv.txt >tmp-mm-9823-linsolv-ans.txt
--- snip ---
#time of linsolv=0.00681901
Time(find_pfaffian_fast)=0.002101 seconds
[3304] P1;
[[0,1],[1/4,-1/6]]

Example of finding a restriction: We consider a left ideal generated by \partial_x (\theta_x + c_{01}-1) - (\theta_x+\theta_y+a) (\theta_x+b_{02}) and \partial_y (\theta_y + c_{11}-1) - (\theta_x+\theta_y+a) (\theta_y+b_{12}) where \theta_x = x \partial_x The Appell function F2 satisfies it. We try to find the restriction to x=0. We input the following commands. 

Ideal=[(-x^2+x)*dx^2+(-y*x*dy+(-a-b_02-1)*x+c_01)*dx-b_02*y*dy-b_02*a,
       (-y*x*dy-b_12*x)*dx+(-y^2+y)*dy^2+((-a-b_12-1)*y+c_11)*dy-b_12*a]$
Xvars=[x,y]$
Std=[(1)*<<0,0>>,(1)*<<0,1>>]$ 
Id = map(dp_ptod,Ideal,poly_dvar(Xvars))$
MData=mt_mm.find_macaulay(Id,Std,Xvars | restriction_var=[x]); 
P2=mt_mm.find_pfaffian(MData,Xvars,2 | use_orig=1);
// Std is a standard basis for the restricted system.

Then, we have the following output. 

[3618] We use a probabilistic method for rank_check with P=65537
--- snip ---
Rank=6
Have allocated 0 M bytes.
0-th rank condition is OK.
We use a probabilistic method to compute the rank of a matrix with entries of rational functions.
The output is a macaulay matrix of degree 1
--- snip ---
[rank=6,[row,col]=[8,6] (size of t(M1)),Indep_cols] is stored in the variable NT_info.
[[[0,0,0,0,0,0,0,c_01],[0,0,0,0,0,0,-y^2+y,0],[0,0,0,0,0,c_01,-b_02*y,0],
  [0,0,0,-y^2+y,0,0,(-a-b_12-3)*y+c_11+1,0],
  [0,0,0,0,c_01+1,(-b_02-1)*y,0,(-b_02-1)*a-b_02-1],
  [0,0,-y^2+y,0,0,(-a-b_12-2)*y+c_11,0,-b_12*a-b_12]],

 [[-b_02*a,-b_02*y],[-b_12*a,(-a-b_12-1)*y+c_11],[0,-b_02*a-b_02],
  [0,(-b_12-1)*a-b_12-1],[0,0],[0,0]],

 [(1)*<<3,0>>,(1)*<<2,1>>,(1)*<<1,2>>,(1)*<<0,3>>,(1)*<<2,0>>,(1)*<<1,1>>,
  (1)*<<0,2>>,(1)*<<1,0>>],
 [(1)*<<0,0>>,(1)*<<0,1>>]]

[3619] [[0,1],[(-b_12*a)/(y^2-y),((-a-b_12-1)*y+c_11)/(y^2-y)]]

An example of finding a restriction on the singular locus: We consider the system of differential equations for the Appell function F_4(a,b,c_{01},c_{11};x,y), which is Id_p below. The curve xy((x-y)^2-2(x+y)+1)=0 is the singular locus of this system and (x,y)=(1/4,1/4) is on the locus. Let F be a holomorphic solution around the point. The values of poly_dact(dx*dy,F,[x,y]),poly_dact(dx^2,F,[x,y]),poly_dact(dy^2,F,[x,y]) (parial derivatives of f) can be obtained from the values of poly_dact(1,F,[x,y]),poly_dact(dx,F,[x,y]),poly_dact(dy,F,[x,y]) (standing for Std) on the singular locus by the Macaulay matrix in MData2 by executing the following codes. 

Id_p=[(-x^2+x)*dx^2+(-2*y*x*dy+(-a-b-1)*x+c_01)*dx-y^2*dy^2+(-a-b-1)*y*dy-b*a,
 -x^2*dx^2+(-2*y*x*dy+(-a-b-1)*x)*dx+(-y^2+y)*dy^2+((-a-b-1)*y+c_11)*dy-b*a];
Xvars=[x,y];
//Restriction at x=y=1/4 included in sing (x-y)^2-2(x+y)+1
Std=map(dp_ptod,[1,dx,dy],[dx,dy])$       // It suceeds at degree 1.
MData=mt_mm.find_macaulay(Id_p,Std,Xvars | restriction_cond=[[x,y],[[x,1/4],[y,1/4]]])$
Args=mt_mm.get_NT_info()[3]$
yang.define_ring(["partial",Xvars])$
MData2=mt_mm.my_macaulay(map(dp_ptod,Id_p,Dvars),Std,Args[3],Args[4]
    | restriction_cond=[[x,y],[]])$

Refer to

@ref{mt_mm.find_pfaffian_fast} @ref{mt_mm.find_pfaffian} mt_mm.find_macaulay

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2.2 mt_mm.my_macaulay

mt_mm.my_macaulay(Id,Std,Deg,N)

:: Multipy all differential operators upto Deg to Id and return a Macaulay matrix.

return

[M1,M2,Ext,Std], M1*Ext+M2*Std=0 modulo Id holds.

Id

A list of generators of the ideal by the distributed polynomial format. Independent variables must be x1, x2, x3, ...

Std

A list of Standard monomials by the distributed polynomial format.

Deg

Degree to generate a Macaulay matrix.

N

N is the number of differential variables = the size of the distributed polynomials=the number of independent variables.

option

restriction_var, see mt_mm.find_macaulay.

option

restriction_cond, see mt_mm.find_macaulay.

Example: Consider the left ideal generated by x_1 \partial_1 - 1/2, x_2 \partial_2 - 1/3. Input the following codes. 

import("mt_mm.rr");
yang.define_ring(["partial",[x1,x2]]);
Id=[x1*<<2,0>>-1/2, x2*<<0,1>>-1/3];
Std=[<<0,0>>,<<1,0>>];
T0=mt_mm.my_macaulay(Id,Std,0,2);
T1=mt_mm.my_macaulay(Id,Std,1,2);

Then, we have 

[3988] T0;
[[[x1,0],[0,x2]],
 [[-1/2,0],[-1/3,0]],
 [(1)*<<2,0>>,(1)*<<0,1>>],
 [(1)*<<0,0>>,(1)*<<1,0>>]]
[3989] T1;
[[[0,0,x1,0,0,0],[0,0,0,0,0,x2],[0,x1,0,0,0,-1/2],[0,0,0,0,x2,2/3],
  [x1,0,1,0,0,0],[0,0,0,x2,0,0]],

 [[-1/2,0],[-1/3,0],[0,0],[0,0],[0,-1/2],[0,-1/3]],

 [(1)*<<3,0>>,(1)*<<2,1>>,(1)*<<2,0>>,(1)*<<1,1>>,(1)*<<0,2>>,(1)*<<0,1>>],
 [(1)*<<0,0>>,(1)*<<1,0>>]]

For example, the 5th rows of T1[0] and T1[1], stands for \partial_1 (x_1 \partial_1^2 - 1/2)= x_1 \partial_1^3 + \partial_1^2 - 1/2 \partial_1.

Refer to

mt_mm.find_macaulay

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3 Invariant subvector space

3.1 mt_mm.ediv_set_field  
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3.1 mt_mm.ediv_set_field

mt_mm.ediv_set_field(Mode)

:: It sets the type of coefficient field.

Mode

When Mode is 1, the elementary divisor is factorized in {\bf Q}[x] (rational number coefficient polynomial ring). Here x can be changed by setting the global variable InvX by calling mt_mm.set_InvX(X). When Mode is 0, the elementary divisor is factorized in {\bf Q}(a,b,\ldots)[x] where a,b, \ldots are parameter variables automatically determined from the input.

Refer to

mt_mm.mat_inv_space

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3.2 mt_mm.mat_inv_space

mt_mm.mat_inv_space(Mat)

:: It returns sets of basis vectors of invariant subvector spaces of Mat

return

a list of sets of basis vectors of invariant subvector spaces by the action of Mat.

Mat

A square matrix.

ediv

Option. When ediv=1, it returns [a list of sets of basis vectors,[ED,L,R]] where ED is the elementary divisor.

Example: We have 3 invariant subvector spaces of the 3 by 3 matrix L below. Then, the matrix can be diagonalized by them. 

[3118] import("invlin0.rr");
Base field is Q.
[3168] B=mt_mm.mat_inv_space(L=[[6,-3,-7],[-1,2,1],[5,-3,-6]]);
[[[ -2/25 2/25 -2/25 ]],[[ 6/25 3/25 3/25 ]],[[ -1/25 0 -1/25 ]]]
[3170] length(B);
3
[3171] L=newmat(3,3,L);
[ 6 -3 -7 ]
[ -1 2 1 ]
[ 5 -3 -6 ]
[3174] P=matrix_transpose(newmat(3,3,[vtol(B[0][0]),vtol(B[1][0]),vtol(B[2][0])]));
[ -2/25 6/25 -1/25 ]
[ 2/25 3/25 0 ]
[ -2/25 3/25 -1/25 ]
[3176] matrix_inverse(P)*L*P;
[ 2 0 0 ]
[ 0 1 0 ]
[ 0 0 -1 ]
[3177] map(print,check5(L))$ // The above can be done by check5.
[[[ -2/25 2/25 -2/25 ]],[[ 6/25 3/25 3/25 ]],[[ -1/25 0 -1/25 ]]]
// invariant subvector spaces.

[ -2/25 6/25 -1/25 ]
[ 2/25 3/25 0 ]
[ -2/25 3/25 -1/25 ]
// transformation matrix

[ 2 0 0 ]
[ 0 1 0 ]
[ 0 0 -1 ]
// diagonalized matrix

[3178] B=mat_inv_space(L | ediv=1)$
[3179] B[1][0];  // The elementary divisor is the diagonal.
[ 1 0 0 ]
[ 0 6/25 0 ]
[ 0 0 x^3-2*x^2-x+2 ]
[3180] fctr(B[1][0][2][2]);
[[1,1],[x-2,1],[x-1,1],[x+1,1]]

Example: The characteristic polynomial is factored into (x^2+1)^2 in {\bf Q}[x]. Then, we obtain two invariant subspaces. 

[3196] B=mt_mm.mat_inv_space([[0,-1,0,0],[1,0,0,0],[0,0,0,1],[0,0,-1,0]]);
[[[ 1 0 0 0 ],[ 0 1 0 0 ]],[[ 0 0 1 0 ],[ 0 0 0 -1 ]]]
[3197] length(B);
2

Example: Block diagonalization. The matrix L is congruent to \pmatrix{1&0&0\cr 0&2&1\cr 0&0&2\cr}. 

[3352] L=matrix_list_to_matrix([[4,0,1],[2,3,2],[0,-2,0]]);
[ 4 0 1 ]
[ 2 3 2 ]
[ 0 -2 0 ]
[3348] B=mt_mm.check5(L);
[[[[ 4 6 -4 ]],[[ -3 -6 4 ],[ -8 -16 12 ]]],[ 4 -3 -8 ]
[ 6 -6 -16 ]
[ -4 4 12 ],[ 3 0 0 ]
[ 0 0 -4 ]
[ 0 1 4 ]]
[3350] B[0];
[[[ 4 6 -4 ]],[[ -3 -6 4 ],[ -8 -16 12 ]]]
[3351] map(length,B[0]);
[1,2]    
//There are one dimensional and two dimensional invariant subspaces for L
[3353] matrix_inverse(B[1])*L*B[1];
[ 3 0 0 ]
[ 0 0 -4 ]
[ 0 1 4 ]
[3354] B[2];
[ 3 0 0 ]
[ 0 0 -4 ]
[ 0 1 4 ]
// the matrix L is block diagonalized as 1 by 1 matrix and 2 by 2 matrix.
[3355]

Example: When the matrix contains parameters, you need to change the base field. L below is congruent to \pmatrix{a&1\cr 0&1\cr}. 

[3110] import("mt_mm.rr");
[3111] mt_mm.ediv_set_field(0);
Base field is Q(params). 
Warning: computation over Q(params) may require huge memory space and time.
0
[3112] B=mt_mm.mat_inv_space(L=[[a-6,-8],[9/2,a+6]]);
[[[ 1 0 ],[ a-6 9/2 ]]]
[3113] length(B);
1   // one invariant subvector space for the linear map L
[3357] B=check5(L);
[[[[ 1 0 ],[ a-6 9/2 ]]],[ 1 a-6 ]
[ 0 9/2 ],[ 0 -a^2 ]
[ 1 2*a ]]
[3359] fctr(matrix_det(B[2]-x*matrix_identity_matrix(2)));
[[1,1],[x-a,2]] // characteristic polynomial is (x-a)^2
Refer to

mt_mm.ediv_set_field

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3.3 mt_mm.is_integral_difference_sol

mt_mm.is_integral_difference_sol(F1,F2)

:: Check if the irreducible polynomials F1 and F2 in x have a common solution with integral difference.

return

If there is a common solution with integral difference, it returns [1,Y] where Y is an integer and x0+Y and x0 are solutions of F1 and F2 respectively. If there is no such solution, it returns 0.

F1

A polynomial in x.

F2

A polynomial in x.

Example: Two examples of irreducible polynomials over Q and Q(a,b). 

[2695] is_integral_difference_sol(x^2+1,(x+@i+1)*(x-@i+1)); 
[1,-1]
[2696] is_integral_difference_sol(x^2+(b/(a+1))^2,(x-(b/(a+1))*@i+3)*(x+(b/(a+1))*@i+3));
[1,3]
Refer to

mt_mm.mat_inv_space @ref{mt_mm.eDiv} @ref{mt_mm.set_InvX}

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4 Moser reduction::

4.1 mt_mm.moser_reduction  
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4.1 mt_mm.moser_reduction

mt_mm.moser_reduction(A,X)

:: It returns the Moser reduced matrix of A with respect to X

return

Moser reduced matrix

A

Coefficient matrix of the ODE dF/dX - AF where F is an unknown vector valued function.

X

Independent variable of the ODE.

Example: the following input outputs the Moser reduced form A2[0] of A. 

import("mt_mm.rr");
A=newmat(4,4,[[-2/x,0,1/(x^2),0],
              [x^2,-(x^2-1)/x,x^2,-x^3],
              [0,x^(-2),x,0],
              [x^2,1/x,0,-(x^2+1)/x]]);
A2=mt_mm.moser_reduction(A,x);


[3405] A2;
[[ (-x^2-1)/(x) x^2 0 x ]
[ 0 (-2)/(x) (1)/(x) 0 ]
[ 0 0 (x^2-1)/(x) (1)/(x) ]
[ -x 1 x (-x^2-1)/(x) ],[ 0 1 0 0 ]
[ 0 0 0 x^2 ]
[ 0 0 x 0 ]
[ 1 0 0 0 ]]
// A2[1] gives the Gauge transformation to obtain the Moser reduced form.
[3406] A2[0]-mt_mm.gauge_transform(A,A2[1],x);
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
Refer to

@ref{mt_mm.gauge_transform}

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5 Restriction::

5.1 mt_mm.restriction_to_pt_  
5.2 mt_mm.v_by_eset  
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5.1 mt_mm.restriction_to_pt_

mt_mm.restriction_to_pt_(Ideal,Gamma,KK,V)

:: It returns the restriction of Ideal to the origin by a probabilistic method.

return

a list R. R[0]: a basis of the restriction. R[1]: column indices for the basis. R[2]: a basis of {\bf C}^{c(\gamma)}. R[3]: echelon form of a matrix for the restriction.

Ideal

Generators of an ideal in the ring of differential operators

Gamma

Approximation parameter \gamma for the restriction. In order to obtain the exact answer, it must be larger than or equal to {\rm max}(s_0,s_1) where s_0 is the maximal non-negative root of b-function and s_1 is the maximal order with respect to (-1,1) weight of the Groebner basis of the Ideal

KK

Approximation parameter k for the restriction. This parameter must be larger than \gamma. If k is sufficiently large, this function returns the exact answer.

Restriction of the system of Appell F1. 

[2077] import("mt_mm.rr");
[3599] Ideal=mt_mm.appell_F1();
  [(-x^2+x)*dx^2+((-y*x+y)*dy+(-a-b1-1)*x+c)*dx-b1*y*dy-b1*a,
   ((-y+1)*x*dy-b2*x)*dx+(-y^2+y)*dy^2+((-a-b2-1)*y+c)*dy-b2*a,
   ((x-y)*dy-b2)*dx+b1*dy]
[3600] Ideal2=base_replace(Ideal,[[a,1/2],[b1,1/3],[b2,1/5],[c,1/7]]);
  [(-x^2+x)*dx^2+((-y*x+y)*dy-11/6*x+1/7)*dx-1/3*y*dy-1/6,
   ((-y+1)*x*dy-1/5*x)*dx+(-y^2+y)*dy^2+(-17/10*y+1/7)*dy-1/10,
   ((x-y)*dy-1/5)*dx+1/3*dy]
[3601] T=mt_mm.restriction_to_pt_(Ideal2,3,4,[x,y] | p=10^10)$
[3602] T[0];
  [1]   // The basis of the restriction at the origin is {1

[3604] Ideal3=base_replace(Ideal2,[[x,x+2],[y,y+3]]);
  [(-x^2-3*x-2)*dx^2+(((-y-3)*x-y-3)*dy-11/6*x-74/21)*dx+(-1/3*y-1)*dy-1/6,
   (((-y-2)*x-2*y-4)*dy-1/5*x-2/5)*dx+(-y^2-5*y-6)*dy^2+(-17/10*y-347/70)*dy-1/10,
   ((x-y-1)*dy-1/5)*dx+1/3*dy]
[3605] T3=mt_mm.restriction_to_pt_(Ideal3,3,4,[x,y] | p=10^10)$
[3606] T3[0];
  [dx,dy,1]   // The basis of the restrition at (x,y)=(2,3)
[3607]

Rational restriction of the system of Appell F2 to x=0. The rank 2 system of ODE is stored in the variable P2. 

import("mt_mm.rr")$
Ideal = [(-x^2+x)*dx^2+(-y*x)*dx*dy+((-a-b1-1)*x+c1)*dx-b1*y*dy-b1*a,
         (-y^2+y)*dy^2+(-x*y)*dy*dx+((-a-b2-1)*y+c2)*dy-b2*x*dx-b2*a]$
Xvars = [x,y]$
//Rule for a probabilistic determination of RStd (Std for the restriction)
Rule=[[y,y+1/3],[a,1/2],[b1,1/3],[b2,1/5],[c1,1/7],[c2,1/11]]$
Ideal_p = base_replace(Ideal,Rule);
RStd=mt_mm.restriction_to_pt_by_linsolv(Ideal_p,Gamma=2,KK=4,[x,y]);
RStd=reverse(map(dp_ptod,RStd[0],[dx,dy]));
Id = map(dp_ptod,Ideal,poly_dvar(Xvars))$
MData = mt_mm.find_macaulay(Id,RStd,Xvars | restriction_var=[x]);
P2 = mt_mm.find_pfaffian(MData,Xvars,2 | use_orig=1);
end$
Refer to

mt_mm.find_macaulay

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5.2 mt_mm.v_by_eset

mt_mm.v_by_eset(L,Eset,V)

:: It returns coefficients vector of L with respect to Eset

return

a list of coefficients of L with respect to Eset.

L

a differential operator.

Eset

a list of monomials of differential variable.

V

a list of variables. 

[1883] import("mt_mm.rr");
[3600] mt_mm.v_by_eset(x*dx+y*dy+a,reverse(mt_mm.eset(2,[x,y])),[x,y]);
  [ 0 0 0 x y a ]

[3601] Eset=mt_mm.eset(2,[x,y]);
  [1,dy,dx,dy^2,dy*dx,dx^2]
[3602] T=mt_mm.dshift_by_eset(x*dx+y*dy+a,Eset,[x,y]);
  // Eset is applied to x*dx+y*dy+a and the result is
 [x*dx+y*dy+a,
  x*dy*dx+y*dy^2+(a+1)*dy,
  x*dx^2+(y*dy+a+1)*dx,
  x*dy^2*dx+y*dy^3+(a+2)*dy^2,
  x*dy*dx^2+(y*dy^2+(a+2)*dy)*dx,
  x*dx^3+(y*dy+a+2)*dx^2]

[3603] T2=map(mt_mm.v_by_eset,T,reverse(mt_mm.eset(3,[x,y])),[x,y]);
[[ 0 0 0 0 0 0 0 x y a ],[ 0 0 0 0 0 x y 0 a+1 0 ],
 [ 0 0 0 0 x y 0 a+1 0 0 ],[ 0 0 x y 0 0 a+2 0 0 0 ],
 [ 0 x y 0 0 a+2 0 0 0 0 ],[ x y 0 0 a+2 0 0 0 0 0 ]]
[3604]
Refer to

mt_mm.restriction_to_pt_

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Index

Jump to:   M  
Index Entry  Section
M
mt_mm.ediv_set_field 3.1 mt_mm.ediv_set_field
mt_mm.find_macaulay 2.1 mt_mm.find_macaulay
mt_mm.is_integral_difference_sol 3.3 mt_mm.is_integral_difference_sol
mt_mm.mat_inv_space 3.2 mt_mm.mat_inv_space
mt_mm.moser_reduction 4.1 mt_mm.moser_reduction
mt_mm.my_macaulay 2.2 mt_mm.my_macaulay
mt_mm.restriction_to_pt_ 5.1 mt_mm.restriction_to_pt_
mt_mm.v_by_eset 5.2 mt_mm.v_by_eset
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