Sm1 OX Server Manual

Edition : auto generated by oxgentexi on 23 February 2009

OpenXM.org

1 SM1 Functions

This chapter describes interface functions for sm1 ox server ox_sm1_forAsir. These interface functions are defined in the file ‘sm1.rr’. The file ‘sm1.rr’ is

at ‘$(OpenXM_HOME)/lib/asir/contrib-packages’. The system sm1 is a system to com-

pute in the ring of differential operators. Many constructions of invariants in the computational algebraic geometry reduce to constructions in the ring of differential operators. Documents on sm1 are in the directory OpenXM/doc/kan96xx.

All the coefficients of input polynomials should be integers for most functions in this section. Other functions accept rational numbers as inputs and it will be explicitely noted in each explanation of these functions.

Let us evaluate the dimensions of the de Rham cohomology groups of X := C \ {0, 1} = C \ V (x(x − 1)). The space X is a two punctured plane, so two loops that encircles the points x = 0 and x = 1 respectively spans the first homology group. Hence, the dimension of the first de Rham cohomology group is 2. sm1 answers the dimensions of the 0th and the first cohomology groups.

     [283] sm1.deRham([x*(x-1),[x]]);
     [1,2]
The author of sm1 : Nobuki Takayama, takayama@math.sci.kobe-u.ac.jp

The author of sm1 packages : Toshinori Oaku, oaku@twcu.ac.jp Reference: [SST] Saito, M., Sturmfels, B., Takayama, N., Grobner Deformations of Hypergeometric Differential Equations, 1999, Springer. http://www.math.kobe-u.ac.jp/KAN

1.1 ox_sm1_forAsir Server

1.1.1 ox_sm1_forAsir
ox_sm1_forAsir

:: sm1 server for asir.

• ox_sm1_forAsir is the sm1 server started from asir by the command sm1.start. In the standard setting,

    ox_sm1_forAsir = ‘$(OpenXM_HOME)/lib/sm1/bin/ox_sm1’ + ‘$(OpenXM_HOME)/lib/sm1/callsm1.sm1’

(macro file)

    + ‘$(OpenXM_HOME)/lib/sm1/callsm1b.sm1’ (macro file)

The macro files ‘callsm1.sm1’ and ‘callsm1b.sm1’ are searched from current

    directory, $(LOAD_SM1_PATH), $(OpenXM_HOME)/lib/sm1, /usr/local/lib/sm1 in

this order.

  • Note for programmers: See the files ‘$(OpenXM_HOME)/src/kxx/oxserver00.c’,
    ‘$(OpenXM_HOME)/src/kxx/sm1stackmachine.c’ to build your own server by reading

sm1 macros.

1.2 Functions

1.2.1 sm1.start
sm1.start()

:: Start ox_sm1_forAsir on the localhost.

return Integer • Start ox_sm1_forAsir on the localhost. It returns the descriptor of ox_sm1_forAsir.

• Set Xm_noX = 1 to start ox_sm1_forAsir without a debug window.

• You might have to set suitable orders of variable by the command ord. For example, when you are working in the ring of differential operators on the variable x and dx (dx stands for ∂/∂x ), sm1 server assumes that the variable dx is collected to the right and the variable x is collected to the left in the printed expression. In the example below, you must not use the variable cc for computation in sm1.

• The variables from a to z except d and o and x0, ..., x20, y0, ..., y20, z0, ..., z20 can be used as variables for ring of differential operators in default. (cf. Sm1_ord_list in

sm1).

• The descriptor is stored in static Sm1_proc. The descriptor can be obtained by the

function sm1.get_Sm1_proc().
[260] ord([da,a,db,b]);
[da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
......... omit ..................
]
[261] a*da;
a*da
[262] cc*dcc;
dcc*cc
[263] sm1.mul(da,a,[a]);
a*da+1
[264] sm1.mul(a,da,[a]);
a*da
Reference
     ox_launch, sm1.push_int0, sm1.push_poly0, ord
1.2.2 sm1.sm1
sm1.sm1(p,s)

:: ask the sm1 server to execute the command string s.

return Void

p Number

s String • It asks the sm1 server of the descriptor number p to execute the command string s. (In

the next example, the descriptor number is 0.)

[261] sm1.sm1(0," ( (x-1)^2 ) . ");
0
[262] ox_pop_string(0);
x^2-2*x+1
[263] sm1.sm1(0," [(x*(x-1))  [(x)]] deRham ");
0
[264] ox_pop_string(0);
[1 , 2]
Reference
     sm1.start, ox_push_int0, sm1.push_poly0, sm1.get_Sm1_proc().
1.2.3 sm1.push_int0
sm1.push_int0(p,f)

:: push the object f to the server with the descriptor number p.

return Void

p Number

f Object

• When type(f) is 2 (recursive polynomial), f is converted to a string (type == 7) and is sent to the server by ox_push_cmo.

• When type(f) is 0 (zero), it is translated to the 32 bit integer zero on the server. Note that ox_push_cmo(p,0) sends CMO_NULL to the server. In other words, the server does not get the 32 bit integer 0 nor the bignum 0.

• sm1 integers are classfied into the 32 bit integer and the bignum. When type(f) is 1 (number), it is translated to the 32 bit integer on the server. Note that ox_push_ cmo(p,1234) send the bignum 1234 to the sm1 server.

• In other cases, ox_push_cmo is called without data conversion.

[219] P=sm1.start();
0
[220] sm1.push_int0(P,x*dx+1);
0
[221] A=ox_pop_cmo(P);
x*dx+1
[223] type(A);
7   (string)
[271] sm1.push_int0(0,[x*(x-1),[x]]);
0
[272] ox_execute_string(0," deRham ");
0
[273] ox_pop_cmo(0);
[1,2]

Reference
          ox_push_cmo
1.2.4 sm1.gb
sm1.gb([f,v,w]|proc=p,sorted=q,dehomogenize=r)

:: computes the Grobner basis of f in the ring of differential operators with the variable v.

sm1.gb_d([f,v,w]|proc=p)

:: computes the Grobner basis of f in the ring of differential operators with the variable v. The result will be returned as a list of distributed polynomials.

return List

p, q, r Number

f, v, w List

• It returns the Grobner basis of the set of polynomials f in the ring of deferential operators with the variables v.

• The weight vectors are given by w, which can be omitted. If w is not given, the graded reverse lexicographic order will be used to compute Grobner basis.

• The return value of sm1.gb is the list of the Grobner basis of f and the initial terms (when w is not given) or initial ideal (when w is given).

• sm1.gb_d returns the results by a list of distributed polynomials. Monomials in each distributed polynomial are ordered in the given order. The return value consists of [variable names, order matrix, grobner basis in districuted polynomials, initial monomials or initial polynomials].

• When a non-term order is given, the Grobner basis is computed in the homogenized Weyl algebra (See Section 1.2 of the book of SST). The homogenization variable h is automatically added.

• When the optional variable q is set, sm1.gb returns, as the third return value, a list of the Grobner basis and the initial ideal with sums of monomials sorted by the given order. Each polynomial is expressed as a string temporally for now. When the optional variable r is set to one, the polynomials are dehomogenized (,i.e., h is set to 1).

     [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
     [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]

In the example above, the set {x∂_{x} + y∂_{y} − 1, y^{2}∂_{y}^{2} + 2} is the Gröbner basis of the input with respect to the graded reverse lexicographic order such that 1 ≤ ∂_{y} ≤ ∂_{x} ≤ y ≤ x ≤ · · ·. The set {x∂_{x}, y^{2}∂_{y}} is the leading monomials (the initial monominals) of the Gröbner basis.

     [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
     [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]

In the example above, two monomials m = x^{a}y^{b}∂_{x}^{c}∂_{y}^{d} and m′ = x^{a′}y^{b′}∂_{x}^{c′}∂_{y}^{d′} are firstly compared by the weight vector (dx,dy,x,y) = (50,2,1,0) (i.e., m is larger than m′ if 50c + 2d + a > 50c′ + 2d′ + a′) and when the comparison is tie, then these are compared by the reverse lexicographic order (i.e., if 50c + 2d + a = 50c′ + 2d′ + a′, then use the reverse lexicogrpahic order).

[294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
      map(print,F[2][0])$
      map(print,F[2][1])$
[595]
   sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
             [x,y],[[dx,1,x,-1],[dy,1]]]);
[[x*dx^2+(y*dy-h^2)*dx-h^3,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx-h^3*dy],
 [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
[596]
   sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
             "x,y",[[dx,1,x,-1],[dy,1]]]);
[[[e0,x,y,H,E,dx,dy,h],
 [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
  [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
  [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
  [0,0,0,0,0,0,0,1]]],
[[(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>+(-1)*
<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0
,0,0,0,1,2>>+(-1)*<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>+(-1)*<<0,0,0,0,0,0
,1,3>>],
 [(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>,(1)*<
<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0,0,0,0,1,2>>+(-1)*<<0,0,0
,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
Reference
     sm1.reduction, sm1.rat_to_p
1.2.5 sm1.deRham
sm1.deRham([f,v]|proc=p)

:: ask the server to evaluate the dimensions of the de Rham cohomology groups of C^n - (the zero set of f =0).

return List

p Number

f String or polynomial

v List

• It returns the dimensions of the de Rham cohomology groups of X = C^n \ V(f ). In other words, it returns [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].

• v is a list of variables. n = length(v).

• sm1.deRham requires huge computer resources. For example, sm1.deRham(0,[x*y*z*(x+y+z-

1)*(x-y),[x,y,z]]) is already very hard.

• To efficiently analyze the roots of b-function, ox_asir should be used from ox_sm1_forAsir. It is recommended to load the communication module for ox_asir by the command

sm1(0,"[(parse) (oxasir.sm1) pushfile] extension"); This command is

automatically executed when ox_sm1_forAsir is started.

• If you make an interruption to the function sm1.deRham by ox_reset(sm1.get_Sm1_ proc());, the server might get out of the standard mode. So, it is strongly recommended to execute the command ox_shutdown(sm1.get_Sm1_proc()); to interrupt and restart the server.

 [332] sm1.deRham([x^3-y^2,[x,y]]);
 [1,1,0]
 [333] sm1.deRham([x*(x-1),[x]]);
 [1,2]
Reference
      sm1.start, deRham (sm1 command)
Algorithm:

Oaku, Takayama, An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation, Journal of pure and applied algebra 139 (1999), 201–233.

1.2.6 sm1.hilbert
sm1.hilbert([f,v]|proc=p)

:: ask the server to compute the Hilbert polynomial for the set of polynomials f.

hilbert_polynomial(f,v)

:: ask the server to compute the Hilbert polynomial for the set of polynomials f.

return Polynomial

p Number

f, v List

• It returns the Hilbert polynomial h(k) of the set of polynomials f with respect to the set of variables v.

• h(k) = dim Q F k/I \cap F k where F k the set of polynomials of which degree is less than or equal to k and I is the ideal generated by the set of polynomials f.

• Note for sm1.hilbert: For an efficient computation, it is preferable that the set of polynomials f is a set of monomials. In fact, this function firstly compute a Grobner basis of f, and then compute the Hilbert polynomial of the initial monomials of the basis. If the input f is already a Grobner basis, a Grobner basis is recomputed in this function, which is a waste of time and Grobner basis computation in the ring of polynomials in sm1 is slower than in asir.

 [346] load("katsura")$

[351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
32
[279] load("katsura")$
[280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
[281] dp_ord();
0
[282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
[(1)*<<1,0,0,0,0,0>>,(1)*<<0,0,0,2,0,0>>,(1)*<<0,0,1,1,0,0>>,(1)*<<0,0,2,0,0,0>>,
 (1)*<<0,1,1,0,0,0>>,(1)*<<0,2,0,0,0,0>>,(1)*<<0,0,0,1,1,1>>,(1)*<<0,0,0,1,2,0>>,
 (1)*<<0,0,1,0,2,0>>,(1)*<<0,1,0,0,2,0>>,(1)*<<0,1,0,1,1,0>>,(1)*<<0,0,0,0,2,2>>,
  (1)*<<0,0,1,0,1,2>>,(1)*<<0,1,0,0,1,2>>,(1)*<<0,1,0,1,0,2>>,(1)*<<0,0,0,0,3,1>>,
  (1)*<<0,0,0,0,4,0>>,(1)*<<0,0,0,0,1,4>>,(1)*<<0,0,0,1,0,4>>,(1)*<<0,0,1,0,0,4>>,
 (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
[283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
[u0,u3^2,u3*u2,u2^2,u2*u1,u1^2,u5*u4*u3,u4^2*u3,u4^2*u2,u4^2*u1,u4*u3*u1,
 u5^2*u4^2,u5^2*u4*u2,u5^2*u4*u1,u5^2*u3*u1,u5*u4^3,u4^4,u5^4*u4,u5^4*u3,
 u5^4*u2,u5^4*u1,u5^6]
[284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
32
Reference
     sm1.start, sm1.gb, longname
1.2.7 sm1.genericAnn
sm1.genericAnn([f,v]|proc=p)

:: It computes the annihilating ideal for f ^s. v is the list of variables. Here, s is v[0] and f is a polynomial in the variables rest(v).

return List

p Number

f Polynomial

v List

• This function computes the annihilating ideal for f ^s. v is the list of variables. Here, s is v[0] and f is a polynomial in the variables rest(v).

[595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
[-x*dx-y*dy-z*dz+3*s,z^2*dy-y^2*dz,z^2*dx-x^2*dz,y^2*dx-x^2*dy]
Reference
     sm1.start
1.2.8 sm1.wTensor0
sm1.wTensor0([f,g,v,w]|proc=p)

:: It computes the D-module theoretic 0-th tensor product of f and g. return List

p Number

f, g, v, w List • It returns the D-module theoretic 0-th tensor product of f and g.

• v is a list of variables. w is a list of weights. The integer w[i] is the weight of the variable v[i].

• sm1.wTensor0 calls wRestriction0 of ox_sm1, which requires a generic weight vector w to compute the restriction. If w is not generic, the computation fails.

• Let F and G be solutions of f and g respectively. Intuitively speaking, the 0-th tensor product is a system of differential equations which annihilates the function FG.

• The answer is a submodule of a free module D^r in general even if the inputs f and g are left ideals of D.

     [258]  sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
     [[-y*x*dx-y*x*dy+4*x+y],[5*x*dx^2+5*x*dx+2*y*dy^2+(-2*y-6)*dy+3],
      [-25*x*dx+(-5*y*x-2*y^2)*dy^2+((5*y+15)*x+2*y^2+16*y)*dy-20*x-8*y-15],
      [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1.2.9 sm1.reduction
sm1.reduction([f,g,v,w]|proc=p)

::

return List

f Polynomial

g, v, w List

p Number (the process number of ox sm1)

• It reduces f by the set of polynomial g in the homogenized Weyl algebra; it applies the division algorithm to f. The set of variables is v and w is weight vectors to determine the order, which can be ommited. sm1.reduction_noH is for the Weyl algebra.

• The return value is of the form [r,c0,[c1,...,cm],[g1,...gm]] where g=[g1, ..., gm] and c0 f + c1 g1 + ... + cm gm = r. r/c0 is the normal form.

• The function reduction reduces reducible terms that appear in lower order terms.

• The functions sm1.reduction d(P,F,G) and sm1.reduction noH d(P,F,G) are for distributed polynomials.

     [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
     [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
     [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
     [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
Reference
          sm1.start, d_true_nf

1.2.10 sm1.xml_tree_to_prefix_string
sm1.xml_tree_to_prefix_string(s|proc=p)

:: Translate OpenMath Tree Expression s in XML to a prefix notation.

return String

p Number

s String • It translate OpenMath Tree Expression s in XML to a prefix notation.

• This function should be moved to om * in a future.

• om_xml_to_cmo(OpenMath Tree Expression) returns CMO TREE. asir has not yet

understood this CMO.

• java execution environment is required. (For example, /usr/local/jdk1.1.8/bin should be in the command search path.)

[263] load("om");
1
[270] F=om_xml(x^4-1);
control: wait OX
Trying to connect to the server... Done.
<OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
<OMS name="times" cd="basic"/><OMA>
<OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
<OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
<OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
<OMI>-1</OMI></OMA></OMA></OMOBJ>
[271] sm1.xml_tree_to_prefix_string(F);
basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
Reference
     om_*, OpenXM/src/OpenMath, eval_str
1.2.11 sm1.syz
sm1.syz([f,v,w]|proc=p)

:: computes the syzygy of f in the ring of differential operators with the variable v.

return List

p Number

f, v, w List

• The return values is of the form [s,[g, m, t]]. Here s is the syzygy of f in the ring of differential operators with the variable v. g is a Groebner basis of f with the weight vector w, and m is a matrix that translates the input matrix f to the Gr\"obner basis g. t is the syzygy of the Gr\"obner basis g. In summary, g = m f and s f = 0 hold as matrices.

• The weight vectors are given by w, which can be omitted. If w is not given, the graded reverse lexicographic order will be used to compute Grobner basis.

• When a non-term order is given, the Grobner basis is computed in the homogenized Weyl algebra (See Section 1.2 of the book of SST). The homogenization variable h is automatically added.

[293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
[[[y*x*dy*dx-2,-x*dx-y*dy+1]],    generators of the syzygy
 [[[x*dx+y*dy-1],[y^2*dy^2+2]],   grobner basis
  [[1,0],[y*dy,-1]],              transformation matrix
 [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
[294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);
[[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
 [[[x^2*dx^2+h^2*x*dx+y^2*dy^2+h^2*y*dy-4*h^4],[y*x*dy*dx-h^4], GB
  [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
 [[1,0],[0,1],[y*dy,-x*dx]],     transformation matrix
 [[y*x*dy*dx-h^4,-x^2*dx^2-h^2*x*dx-y^2*dy^2-h^2*y*dy+4*h^4]]]]
1.2.12 sm1.mul
sm1.mul(f,g,v|proc=p)

:: ask the sm1 server to multiply f and g in the ring of differential operators over v.

return Polynomial or List

p Number

f, g Polynomial or List

v List • Ask the sm1 server to multiply f and g in the ring of differential operators over v.

• sm1.mul_h is for homogenized Weyl algebra.

• BUG: sm1.mul(p0*dp0,1,[p0]) returns dp0*p0+1. A variable order such that dvariables come after non-d-variables is necessary for the correct computation.

[277] sm1.mul(dx,x,[x]);
x*dx+1
[278] sm1.mul([x,y],[1,2],[x,y]);
x+2*y
[279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
[[x+2,y+4],[3*x+4,3*y+8]]
1.2.13 sm1.distraction
sm1.distraction([f,v,x,d,s]|proc=p)

:: ask the sm1 server to compute the distraction of f.

return List p Number

f Polynomial

v,x,d,s List

• It asks the sm1 server of the descriptor number p to compute the distraction of f in the ring of differential operators with variables v.

• x is a list of x-variables and d is that of d-variables to be distracted. s is a list of variables to express the distracted f.

• Distraction is roughly speaking to replace x*dx by a single variable x. See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.

[280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
x
[281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
x^2-x
[282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
x^2+3*x+2
[283] fctr(@);
[[1,1],[x+1,1],[x+2,1]]
[284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
(x^2-x)*dy+x*y
Reference
     distraction2(sm1),

1.2.14 sm1.gkz
sm1.gkz([A,B]|proc=p)

:: Returns the GKZ system (A-hypergeometric system) associated to the matrix A with the parameter vector B.

return List

p Number

A, B List

• Returns the GKZ hypergeometric system (A-hypergeometric system) associated to the matrix

[280] sm1.gkz([  [[1,1,1,1],[0,1,3,4]],  [0,2] ]);
[[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
 -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
 [x1,x2,x3,x4]]

1.2.15 sm1.mgkz
sm1.mgkz([A,W,B]|proc=p)

:: Returns the modified GKZ system (A-hypergeometric system) associated to the matrix A and the weight w with the parameter vector B.

return List

p Number

A, W, B List

• Returns the modified GKZ hypergeometric system (A-hypergeometric system) associated to the matrix

• http://arxiv.org/abs/0707.0043

     [280] sm1.mgkz([ [[1,2,3]], [1,2,1], [a/2]]);
     [[6*x3*dx3+4*x2*dx2+2*x1*dx1-a,-x4*dx4+x3*dx3+2*x2*dx2+x1*dx1,
       -dx2+dx1^2,-x4^2*dx3+dx1*dx2],[x1,x2,x3,x4]]
     Modified A-hypergeometric system for
     A=(1,2,3), w=(1,2,1), beta=(a/2).
1.2.16 sm1.appell1
sm1.appell1(a|proc=p)

:: Returns the Appell hypergeometric system F 1 or F D.

return List

p Number

a List

• Returns the hypergeometric system for the Lauricella function F D(a,b1,b2,...,bn,c;x1,...,xn) where a =(a,c,b1,...,bn). When n=2, the Lauricella function is called the Appell function F 1. The parameters a, c, b1, ..., bn may be rational numbers.

• It does not call sm1 function appell1. As a concequence, when parameters are rational or symbolic, this function also works as well as integral parameters.

     [281] sm1.appell1([1,2,3,4]);
     [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
       (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
       ((-x2+x1)*dx1+3)*dx2-4*dx1],       equations
      [x1,x2]]                            the list of variables
     [282] sm1.gb(@);
     [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
       +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
       +(-4*x2-4*x1)*dx1-4,

       (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
      +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
      [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
     [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
     3
     [285] Mu=2$ Beta = 1/3$
     [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
     4

1.2.17 sm1.appell4
sm1.appell4(a|proc=p)

:: Returns the Appell hypergeometric system F 4 or F C.

return List

p Number

a List

• Returns the hypergeometric system for the Lauricella function F 4(a,b,c1,c2,...,cn;x1,...,xn) where a =(a,b,c1,...,cn). When n=2, the Lauricella function is called the Appell function F 4. The parameters a, b, c1, ..., cn may be rational numbers.

•

• It does not call sm1 function appell4. As a concequence, when parameters are rational or symbolic, this function also works as well as integral parameters.

     [281] sm1.appell4([1,2,3,4]);
       [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
       (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
                                                                   equations
         [x1,x2]]                                      the list of variables
     [282] sm1.rank(@);
     4

1.2.18 sm1.rank
sm1.rank(a|proc=p)

:: Returns the holonomic rank of the system of differential equations a.

return Number

p Number a List

• It evaluates the dimension of the space of holomorphic solutions at a generic point of the system of differential equations a. The dimension is called the holonomic rank.

• a is a list consisting of a list of differential equations and a list of variables.

• sm1.rrank returns the holonomic rank when a is regular holonomic. It is generally faster than sm1.rank.

     [284]  sm1.gkz([  [[1,1,1,1],[0,1,3,4]],  [0,2] ]);
     [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
       -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
      [x1,x2,x3,x4]]
     [285] sm1.rrank(@);
     4
     [286]  sm1.gkz([  [[1,1,1,1],[0,1,3,4]],  [1,2]]);
     [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
      -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
      [x1,x2,x3,x4]]
     [287] sm1.rrank(@);
     5

1.2.19 sm1.auto_reduce
sm1.auto_reduce(s|proc=p)

:: Set the flag "AutoReduce" to s.

return Number

p Number

s Number • If s is 1, then all Grobner bases to be computed will be the reduced Grobner bases.

• If s is 0, then Grobner bases are not necessarily the reduced Grobner bases. This is the default.

1.2.20 sm1.slope
sm1.slope(ii,v,f_filtration,v_filtration|proc=p)

:: Returns the slopes of differential equations ii.

return List

p Number

ii List (equations)

v List (variables)

f filtration List (weight vector) v filtration

List (weight vector)

• sm1.slope returns the (geometric) slopes of the system of differential equations ii along the hyperplane specified by the V filtration v filtration.

• v is a list of variables.

• The return value is a list of lists. The first entry of each list is the slope and the second entry is the weight vector for which the microcharacteristic variety is not bihomogeneous.

Algorithm: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger, How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996" Note that the signs of the slopes are negative, but the absolute values of the slopes are returned.

     [284] A= sm1.gkz([  [[1,2,3]],  [-3] ]);

     [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
     [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
          (* This is an interesting example given by Laura Matusevich,
             June 9, 2001 *)
     [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);

Reference
          sm.gb
1.2.21 sm1.ahg
sm1.ahg(A)

: It idential with sm1.gkz(A).

1.2.22 sm1.bfunction
sm1.bfunction(F)

: It computes the global b-function of F.

Description:

It no longer calls sm1’s original bfunction. Instead, it calls asir "bfct". Algorithm:

M.Noro, Mathematical Software, icms 2002, pp.147–157. Example:

      sm1.bfunction(x^2-y^3);

1.2.23 sm1.call_sm1
sm1.call_sm1(F)

: It executes F on the sm1 server. See also sm1.

1.2.24 sm1.ecart_homogenize01Ideal
sm1.ecart_homogenize01Ideal(A)

: It (0,1)-homogenizes the ideal A[0]. Note that it is not an elementwise homogenization.

Example:

input1
F=[(1-x)*dx+1]$ FF=[F,"x,y"]$
sm1.ecart_homogenize01Ideal(FF);
intput2
F=sm1.appell1([1,2,3,4]);
sm1.ecart_homogenize01Ideal(F);

1.2.25 sm1.ecartd_gb
sm1.ecartd_gb(A)

: It returns a standard basis of A by using a tangent cone algorithm. h[0,1](D)- homogenization is used. If the option rv="dp" (return value="dp") is given, the answer is returned in distributed polynomials.

Example:

input1
F=[2*(1-x-y)*dx+1,2*(1-x-y)*dy+1]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
sm1.ecartd_gb(FF);
output1
[[(-2*x-2*y+2)*dx+h,(-2*x-2*y+2)*dy+h],[(-2*x-2*y+2)*dx,(-2*x-2*y+2)*dy]]
input2
F=[2*(1-x-y)*dx+h,2*(1-x-y)*dy+h]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1,dx,1,dy,1]],["noAutoHomogenize",1]]$
sm1.ecartd_gb(FF);
1.2.26 sm1.ecartd_gb_oxRingStructure
sm1.ecartd_gb_oxRingStructure()

: It returns the oxRingStructure of the most recent ecartd gb computation.

1.2.27 sm1.ecartd_isSameIdeal_h
sm1.ecartd_isSameIdeal_h(F)

: Here, F=[II,JJ,V]. It compares two ideals II and JJ in h[0,1](D) alg.

Example:

input
II=[(1-x)^2*dx+h*(1-x)]$ JJ = [(1-x)*dx+h]$
V=[x]$
sm1.ecartd_isSameIdeal_h([II,JJ,V]);
1.2.28 sm1.ecartd_reduction
sm1.ecartd_reduction(F,A)

: It returns a reduced form of F in terms of A by using a tangent cone algorithm. h[0,1](D)-homogenization is used.

Example:

input
F=[2*(1-x-y)*dx+h,2*(1-x-y)*dy+h]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
sm1.ecartd_reduction(dx+dy,FF);
1.2.29 sm1.ecartd_reduction_noh
sm1.ecartd_reduction_noh(F,A)

: It returns a reduced form of F in terms of A by using a tangent cone algorithm. h[0,1](D)-homogenization is NOT used. A[0] must not contain the variable h.

Example:

 F=[2*(1-x-y)*dx+1,2*(1-x-y)*dy+1]$
   FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
   sm1.ecartd_reduction_noh(dx+dy,FF);
1.2.30 sm1.ecartd_syz
sm1.ecartd_syz(A)

: It returns a syzygy of A by using a tangent cone algorithm. h[0,1](D)- homogenization is used. If the option rv="dp" (return value="dp") is given, the answer is returned in distributed polynomials. The return value is in the format [s,[g,m,t]]. s is the generator of the syzygies, g is the Grobner basis, m is the translation matrix from the generators to g. t is the syzygy of g.

Example:

input1
F=[2*(1-x-y)*dx+1,2*(1-x-y)*dy+1]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
sm1.ecartd_syz(FF);
input2
F=[2*(1-x-y)*dx+h,2*(1-x-y)*dy+h]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1,dx,1,dy,1]],["noAutoHomogenize",1]]$
sm1.ecartd_syz(FF);
1.2.31 sm1.gb_oxRingStructure
sm1.gb_oxRingStructure()

: It returns the oxRingStructure of the most recent gb computation.

1.2.32 sm1.gb_reduction
sm1.gb_reduction(F,A)

: It returns a reduced form of F in terms of A by using a normal form algorithm.

h[1,1](D)-homogenization is used.

Example:

input
F=[2*(h-x-y)*dx+h^2,2*(h-x-y)*dy+h^2]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1,dx,1,dy,1]]]$
sm1.gb_reduction((h-x-y)^2*dx*dy,FF);
1.2.33 sm1.gb_reduction_noh
sm1.gb_reduction_noh(F,A)

: It returns a reduced form of F in terms of A by using a normal form algorithm.

Example:

input
F=[2*dx+1,2*dy+1]$
FF=[F,"x,y",[[dx,1,dy,1]]]$
sm1.gb_reduction_noh((1-x-y)^2*dx*dy,FF);
1.2.34 sm1.generalized_bfunction
sm1.generalized_bfunction(I,V,VD,W)

: It computes the generalized b-function (indicial equation) of I with respect to the weight W.

Description:

It no longer calls sm1’s original function. Instead, it calls asir "generic bfct". Example:

sm1.generalized_bfunction([x^2*dx^2-1/2,dy^2],[x,y],[dx,dy],[-1,0,1,0]);
1.2.35 sm1.isSameIdeal_in_Dalg
sm1.isSameIdeal_in_Dalg(I,J,V)

: It compares two ideals I and J in D alg (algebraic D with variables V, no homogenization).

Example:

Input1
II=[(1-x)^2*dx+(1-x)]$ JJ = [(1-x)*dx+1]$ V=[x]$
sm1.isSameIdeal_in_Dalg(II,JJ,V);
1.2.36 sm1.laplace
sm1.laplace(L,V,VL)

: It returns the Laplace transformation of L for VL. V is the list of space variables. The numbers in coefficients must not be rational with a non-1 denominator. cf. ptozp

Example:

L1=sm1.laplace(dt-(3*t^2-x),[x,t],[t,dt]);
L2=sm1.laplace(dx+t,[x,t],[t,dt]);
sm1.restriction([L1,L2],[t,x],[t] | degree=0);
1.2.37 sm1.restriction
sm1.restriction(I,V,R)

: It computes the restriction of I as a D-module to the set defined by R. V is the list of variables. When the optional variable degree=d is given, only the restrictions from 0 to d are computed. Note that, in case of vector input, RESTRICTION VARIABLES MUST APPEAR FIRST in the list of variable V. We are using wbfRoots to get the roots of b-functions, so we can use only generic weight vector for now.

sm1.restriction(I,V,R | degree=key0)

: This function allows optional variables degree

Algorithm:

T.Oaku and N.Takayama, math.AG/9805006, http://xxx.langl.gov Example:

sm1.restriction([dx^2-x,dy^2-1],[x,y],[y]);

1.2.38 sm1.saturation
sm1.saturation(T)

: T = [I,J,V]. It returns saturation of I with respect to J^infty. V is a list of variables.

Example:

      sm1.saturation([[x2^2,x2*x4, x2, x4^2], [x2,x4], [x2,x4]]);
1.2.39 sm1.ahg
sm1.ahg(A)

: It idential with sm1.gkz(A).

1.2.40 sm1.bfunction
sm1.bfunction(F)

: It computes the global b-function of F.

Description:

It no longer calls sm1’s original bfunction. Instead, it calls asir "bfct". Algorithm:

M.Noro, Mathematical Software, icms 2002, pp.147–157. Example:

      sm1.bfunction(x^2-y^3);
1.2.41 sm1.call_sm1
sm1.call_sm1(F)

: It executes F on the sm1 server. See also sm1.

1.2.42 sm1.ecart_homogenize01Ideal
sm1.ecart_homogenize01Ideal(A)

: It (0,1)-homogenizes the ideal A[0]. Note that it is not an elementwise homogenization.

Example:

      input1
        F=[(1-x)*dx+1]$ FF=[F,"x,y"]$
        sm1.ecart_homogenize01Ideal(FF);
      intput2
        F=sm1.appell1([1,2,3,4]);
        sm1.ecart_homogenize01Ideal(F);

1.2.43 sm1.ecartd_gb
sm1.ecartd_gb(A)

: It returns a standard basis of A by using a tangent cone algorithm. h[0,1](D)- homogenization is used. If the option rv="dp" (return value="dp") is given, the answer is returned in distributed polynomials.

Example:

input1
F=[2*(1-x-y)*dx+1,2*(1-x-y)*dy+1]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
sm1.ecartd_gb(FF);
output1
[[(-2*x-2*y+2)*dx+h,(-2*x-2*y+2)*dy+h],[(-2*x-2*y+2)*dx,(-2*x-2*y+2)*dy]]
input2
F=[2*(1-x-y)*dx+h,2*(1-x-y)*dy+h]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1,dx,1,dy,1]],["noAutoHomogenize",1]]$
sm1.ecartd_gb(FF);
1.2.44 sm1.ecartd_gb_oxRingStructure
sm1.ecartd_gb_oxRingStructure()

: It returns the oxRingStructure of the most recent ecartd gb computation.

1.2.45 sm1.ecartd_isSameIdeal_h
sm1.ecartd_isSameIdeal_h(F)

: Here, F=[II,JJ,V]. It compares two ideals II and JJ in h[0,1](D) alg.

Example:

input
II=[(1-x)^2*dx+h*(1-x)]$ JJ = [(1-x)*dx+h]$
V=[x]$
sm1.ecartd_isSameIdeal_h([II,JJ,V]);
1.2.46 sm1.ecartd_reduction
sm1.ecartd_reduction(F,A)

: It returns a reduced form of F in terms of A by using a tangent cone algorithm.

h[0,1](D)-homogenization is used.

Example:

input
F=[2*(1-x-y)*dx+h,2*(1-x-y)*dy+h]$

FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
sm1.ecartd_reduction(dx+dy,FF);
1.2.47 sm1.ecartd_reduction_noh
sm1.ecartd_reduction_noh(F,A)

: It returns a reduced form of F in terms of A by using a tangent cone algorithm. h[0,1](D)-homogenization is NOT used. A[0] must not contain the variable h.

Example:

 F=[2*(1-x-y)*dx+1,2*(1-x-y)*dy+1]$
   FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
   sm1.ecartd_reduction_noh(dx+dy,FF);
1.2.48 sm1.ecartd_syz
sm1.ecartd_syz(A)

: It returns a syzygy of A by using a tangent cone algorithm. h[0,1](D)- homogenization is used. If the option rv="dp" (return value="dp") is given, the answer is returned in distributed polynomials. The return value is in the format [s,[g,m,t]]. s is the generator of the syzygies, g is the Grobner basis, m is the translation matrix from the generators to g. t is the syzygy of g.

Example:

input1
F=[2*(1-x-y)*dx+1,2*(1-x-y)*dy+1]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1]]]$
sm1.ecartd_syz(FF);
input2
F=[2*(1-x-y)*dx+h,2*(1-x-y)*dy+h]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1,dx,1,dy,1]],["noAutoHomogenize",1]]$
sm1.ecartd_syz(FF);
1.2.49 sm1.gb_oxRingStructure
sm1.gb_oxRingStructure()

: It returns the oxRingStructure of the most recent gb computation.

1.2.50 sm1.gb_reduction
sm1.gb_reduction(F,A)

: It returns a reduced form of F in terms of A by using a normal form algorithm. h[1,1](D)-homogenization is used.

Example:

input
F=[2*(h-x-y)*dx+h^2,2*(h-x-y)*dy+h^2]$
FF=[F,"x,y",[[dx,1,dy,1],[x,-1,y,-1,dx,1,dy,1]]]$
sm1.gb_reduction((h-x-y)^2*dx*dy,FF);
1.2.51 sm1.gb_reduction_noh
sm1.gb_reduction_noh(F,A)

: It returns a reduced form of F in terms of A by using a normal form algorithm.

Example:

input
F=[2*dx+1,2*dy+1]$
FF=[F,"x,y",[[dx,1,dy,1]]]$
sm1.gb_reduction_noh((1-x-y)^2*dx*dy,FF);
1.2.52 sm1.generalized_bfunction
sm1.generalized_bfunction(I,V,VD,W)

: It computes the generalized b-function (indicial equation) of I with respect to the weight W.

Description:

It no longer calls sm1’s original function. Instead, it calls asir "generic bfct". Example:

sm1.generalized_bfunction([x^2*dx^2-1/2,dy^2],[x,y],[dx,dy],[-1,0,1,0]);
1.2.53 sm1.isSameIdeal_in_Dalg
sm1.isSameIdeal_in_Dalg(I,J,V)

: It compares two ideals I and J in D alg (algebraic D with variables V, no homogenization).

Example:

Input1
II=[(1-x)^2*dx+(1-x)]$ JJ = [(1-x)*dx+1]$ V=[x]$
sm1.isSameIdeal_in_Dalg(II,JJ,V);
1.2.54 sm1.laplace
sm1.laplace(L,V,VL)

: It returns the Laplace transformation of L for VL. V is the list of space variables. The numbers in coefficients must not be rational with a non-1 denominator. cf. ptozp Example:

L1=sm1.laplace(dt-(3*t^2-x),[x,t],[t,dt]);
L2=sm1.laplace(dx+t,[x,t],[t,dt]);
sm1.restriction([L1,L2],[t,x],[t] | degree=0);
1.2.55 sm1.restriction
sm1.restriction(I,V,R)

: It computes the restriction of I as a D-module to the set defined by R. V is the list of variables. When the optional variable degree=d is given, only the restrictions from 0 to d are computed. Note that, in case of vector input, RESTRICTION VARIABLES MUST APPEAR FIRST in the list of variable V. We are using wbfRoots to get the roots of b-functions, so we can use only generic weight vector for now.

sm1.restriction(I,V,R | degree=key0)

: This function allows optional variables degree

Algorithm:

T.Oaku and N.Takayama, math.AG/9805006, http://xxx.langl.gov Example:

sm1.restriction([dx^2-x,dy^2-1],[x,y],[y]);
1.2.56 sm1.saturation
sm1.saturation(T)

: T = [I,J,V]. It returns saturation of I with respect to J^infty. V is a list of variables.

Example:

sm1.saturation([[x2^2,x2*x4, x2, x4^2], [x2,x4], [x2,x4]]);

Index 25

Index

(Index is nonexistent) (Index is nonexistent)

Short Contents

1 SM1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Table of Contents

1 SM1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 ox_sm1_forAsir Server . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 ox_sm1_forAsir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 sm1.start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.2 sm1.sm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.3 sm1.push_int0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.4 sm1.gb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.5 sm1.deRham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.6 sm1.hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.7 sm1.genericAnn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.8 sm1.wTensor0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.9 sm1.reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.10 sm1.xml_tree_to_prefix_string . . . . . . . . . . . . . . 8

1.2.11 sm1.syz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.12 sm1.mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.13 sm1.distraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.14 sm1.gkz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.15 sm1.mgkz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.16 sm1.appell1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.17 sm1.appell4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.18 sm1.rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.19 sm1.auto_reduce . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.20 sm1.slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.21 sm1.ahg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.22 sm1.bfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.23 sm1.call_sm1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.24 sm1.ecart_homogenize01Ideal . . . . . . . . . . . . . . . 16

1.2.25 sm1.ecartd_gb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.26 sm1.ecartd_gb_oxRingStructure . . . . . . . . . . . . . 16

1.2.27 sm1.ecartd_isSameIdeal_h . . . . . . . . . . . . . . . . . . 16 1.2.28 sm1.ecartd_reduction . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.29 sm1.ecartd_reduction_noh . . . . . . . . . . . . . . . . . . 17 1.2.30 sm1.ecartd_syz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.31 sm1.gb_oxRingStructure. . . . . . . . . . . . . . . . . . . . . 18 1.2.32 sm1.gb_reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.33 sm1.gb_reduction_noh . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.34 sm1.generalized_bfunction . . . . . . . . . . . . . . . . . 18

1.2.35 sm1.isSameIdeal_in_Dalg . . . . . . . . . . . . . . . . . . . 19 1.2.36 sm1.laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.37 sm1.restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.38 sm1.saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.39 sm1.ahg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.40 sm1.bfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.41 sm1.call_sm1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.42 sm1.ecart_homogenize01Ideal . . . . . . . . . . . . . . . 20

1.2.43 sm1.ecartd_gb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.44 sm1.ecartd_gb_oxRingStructure . . . . . . . . . . . . . 21

1.2.45 sm1.ecartd_isSameIdeal_h . . . . . . . . . . . . . . . . . . 21 1.2.46 sm1.ecartd_reduction . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.47 sm1.ecartd_reduction_noh . . . . . . . . . . . . . . . . . . 22 1.2.48 sm1.ecartd_syz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.49 sm1.gb_oxRingStructure. . . . . . . . . . . . . . . . . . . . . 22 1.2.50 sm1.gb_reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.51 sm1.gb_reduction_noh . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.52 sm1.generalized_bfunction . . . . . . . . . . . . . . . . . 23

1.2.53 sm1.isSameIdeal_in_Dalg . . . . . . . . . . . . . . . . . . . 23 1.2.54 sm1.laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.55 sm1.restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.2.56 sm1.saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25