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version 1.6, 2002/08/23 08:16:13

version 1.16, 2004/03/05 19:05:11

Line 1

/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.5 2002/08/11 08:39:47 takayama Exp $ */

/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.15 2004/03/05 15:56:40 ohara Exp $ */

/*&C-texi

/*&C

@c DO NOT EDIT THIS FILE oxphc.texi

/*&C-texi

/*&C

@node SM1 Functions,,, Top

/*&jp-texi

/*&ja

@chapter SM1 $BH!?t(B

$B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}

Line 32 $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE

Line 33 $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE

$B<!85$rEz$($k(B.

@end tex

/*&eg-texi

/*&en

@chapter SM1 Functions

This chapter describes interface functions for

Line 67 Hence, the dimension of the first de Rham cohomology g

Line 68 Hence, the dimension of the first de Rham cohomology g

cohomology groups.

@end tex

/*&C-texi

/*&C

@example

@include opening.texi

[283] sm1_deRham([x*(x-1),[x]]);

[283] sm1.deRham([x*(x-1),[x]]);

[1,2]

@end example

/*&C-texi

/*&C

@noindent

The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*

The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*

Line 86 Grobner Deformations of Hypergeometric Differential Eq

Line 87 Grobner Deformations of Hypergeometric Differential Eq

See the appendix.

/*&C

@menu

* ox_sm1_forAsir::

* sm1_start::

* sm1.start::

* sm1::

* sm1.sm1::

* sm1_push_int0::

* sm1.push_int0::

* sm1_gb::

* sm1.gb::

* sm1_deRham::

* sm1.deRham::

* sm1_hilbert::

* sm1.hilbert::

* hilbert_polynomial::

* sm1.genericAnn::

* sm1_genericAnn::

* sm1.wTensor0::

* sm1_wTensor0::

* sm1.reduction::

* sm1_reduction::

* sm1.xml_tree_to_prefix_string::

* sm1_xml_tree_to_prefix_string::

* sm1.syz::

* sm1_syz::

* sm1.mul::

* sm1_mul::

* sm1.distraction::

* sm1_distraction::

* sm1.gkz::

* sm1_gkz::

* sm1.appell1::

* sm1_appell1::

* sm1.appell4::

* sm1_appell4::

* sm1.rank::

* sm1_rank::

* sm1.auto_reduce::

* sm1_auto_reduce::

* sm1.slope::

* sm1_slope::

* sm1.ahg::

* sm1.bfunction::

* sm1.generalized_bfunction::

* sm1.restriction::

* sm1.saturation::

@end menu

/*&jp-texi

/*&ja

@section @code{ox_sm1_forAsir} $B%5!<%P(B

/*&eg-texi

/*&en

@section @code{ox_sm1_forAsir} Server

/*&eg-texi

/*&en

@node ox_sm1_forAsir,,, Top

@node ox_sm1_forAsir,,, SM1 Functions

@subsection @code{ox_sm1_forAsir}

@findex ox_sm1_forAsir

@table @t

Line 130 See the appendix.

Line 135 See the appendix.

@itemize @bullet

@item

@code{ox_sm1_forAsir} is the @code{sm1} server started from asir

by the command @code{sm1_start}.

by the command @code{sm1.start}.

In the standard setting, @*

@code{ox_sm1_forAsir} =

@file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}

Line 150 See the appendix.

Line 155 See the appendix.

to build your own server by reading @code{sm1} macros.

@end itemize

/*&jp-texi

/*&ja

@node ox_sm1_forAsir,,, Top

@node ox_sm1_forAsir,,, SM1 Functions

@subsection @code{ox_sm1_forAsir}

@findex ox_sm1_forAsir

@table @t

Line 161 to build your own server by reading @code{sm1} macros.

Line 166 to build your own server by reading @code{sm1} macros.

@itemize @bullet

@item

$B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B

@code{sm1_start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.

@code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.

$BI8=`E*@_Dj$G$O(B, @*

@code{ox_sm1_forAsir} =

Line 183 to build your own server by reading @code{sm1} macros.

Line 188 to build your own server by reading @code{sm1} macros.

@end itemize

def sm1_check_server(P) {

M=ox_get_serverinfo(P);

if (M == []) {

return(sm1_start());

}

if (M[0][1] != "Ox_system=ox_sm1_ox_sm1_forAsir") {

print("Warning: the server number ",0)$

print(P,0)$

print(" is not ox_sm1_forAsir server.")$

print("Starting ox_sm1_forAsir server on the localhost.")$

return(sm1_start());

}

return(P);

}

/*&jp-texi

/*&ja

@section $BH!?t0lMw(B

/*&eg-texi

/*&en

@section Functions

/*&eg-texi

/*&en

@c sort-sm1_start

@c sort-sm1.start

@node sm1_start,,, SM1 Functions

@node sm1.start,,, SM1 Functions

@subsection @code{sm1_start}

@subsection @code{sm1.start}

@findex sm1_start

@findex sm1.start

@table @t

@item sm1_start()

@item sm1.start()

:: Start @code{ox_sm1_forAsir} on the localhost.

@end table

Line 242 for computation in @code{sm1}.

Line 233 for computation in @code{sm1}.

and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},

@code{z0}, ..., @code{z20} can be used as variables for ring of

differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).

@item The descriptor is stored in @code{Sm1_proc}.

@item The descriptor is stored in @code{static Sm1_proc}.

The descriptor can be obtained by the function

@code{sm1.get_Sm1_proc()}.

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_start

@c sort-sm1.start

@node sm1_start,,, SM1 Functions

@node sm1.start,,, SM1 Functions

@subsection @code{sm1_start}

@subsection @code{sm1.start}

@findex sm1_start

@findex sm1.start

@table @t

@item sm1_start()

@item sm1.start()

:: localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.

@end table

Line 282 differential operators in default. (cf. @code{Sm1_ord_

Line 275 differential operators in default. (cf. @code{Sm1_ord_

$B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},

@code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B

$B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).

@item $B<1JLHV9f$O(B @code{Sm1_proc} $B$K3JG<$5$l$k(B.

@item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.

$B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.

@end itemize

/*&C-texi

/*&C

@example

[260] ord([da,a,db,b]);

[da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,

Line 295 differential operators in default. (cf. @code{Sm1_ord_

Line 289 differential operators in default. (cf. @code{Sm1_ord_

a*da

[262] cc*dcc;

dcc*cc

[263] sm1_mul(da,a,[a]);

[263] sm1.mul(da,a,[a]);

a*da+1

[264] sm1_mul(a,da,[a]);

[264] sm1.mul(a,da,[a]);

a*da

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{ox_launch}, @code{sm1_push_int0}, @code{sm1_push_poly0},

@code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},

@code{ord}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{ox_launch}, @code{sm1_push_int0}, @code{sm1_push_poly0},

@code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},

@code{ord}

@end table

def sm1_start() {

extern Sm1_lib;

extern Xm_noX;

extern Sm1_proc;

if (Xm_noX) {

P = ox_launch_nox(0,Sm1_lib+"/bin/ox_sm1_forAsir");

}else{

P = ox_launch(0,Sm1_lib+"/bin/ox_sm1_forAsir");

}

if (Xm_noX) {

sm1(P," oxNoX ");

}

ox_check_errors(P);

Sm1_proc = P;

return(P);

}

/*&en

/* ox_sm1 */

/* P is the process number */

def sm1flush(P) {

ox_execute_string(P,"[(flush)] extension pop");

}

def sm1push(P,F) {

G = ox_ptod(F);

ox_push_cmo(P,G);

}

/*&eg-texi

@c sort-sm1

@node sm1,,, SM1 Functions

@node sm1.sm1,,, SM1 Functions

@subsection @code{sm1}

@subsection @code{sm1.sm1}

@findex sm1

@findex sm1.sm1

@table @t

@item sm1(@var{p},@var{s})

@item sm1.sm1(@var{p},@var{s})

:: ask the @code{sm1} server to execute the command string @var{s}.

@end table

Line 368 String

Line 334 String

@itemize @bullet

@item It asks the @code{sm1} server of the descriptor number @var{p}

to execute the command string @var{s}.

(In the next example, the descriptor number is 0.)

@end itemize

/*&jp-texi

/*&ja

@node sm1,,, SM1 Functions

@node sm1.sm1,,, SM1 Functions

@subsection @code{sm1}

@subsection @code{sm1.sm1}

@findex sm1

@findex sm1.sm1

@table @t

@item sm1(@var{p},@var{s})

@item sm1.sm1(@var{p},@var{s})

:: $B%5!<%P(B @code{sm1} $B$K%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$K$?$N$`(B.

@end table

Line 391 to execute the command string @var{s}.

Line 358 to execute the command string @var{s}.

@itemize @bullet

@item $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B

$B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.

($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)

@end itemize

/*&C-texi

/*&C

@example

[261] sm1(0," ( (x-1)^2 ) . ");

[261] sm1.sm1(0," ( (x-1)^2 ) . ");

[262] ox_pop_string(0);

x^2-2*x+1

[263] sm1(0," [(x*(x-1)) [(x)]] deRham ");

[263] sm1.sm1(0," [(x*(x-1)) [(x)]] deRham ");

[264] ox_pop_string(0);

[1 , 2]

@end example

def sm1(P,F) {

ox_execute_string(P,F);

/*&ja

sm1flush(P);

}

/*&jp-texi

@table @t

@item $B;2>H(B

@code{sm1_start}, @code{ox_push_int0}, @code{sm1_push_poly0}, @code{Sm1_proc}.

@code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.

@end table

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm1_start}, @code{ox_push_int0}, @code{sm1_push_poly0}, @code{Sm1_proc}.

@code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.

@end table

def sm1pop(P) {

return(ox_pop_cmo(P));

}

def sm1_to_asir_form(V) { return(toAsirForm(V)); }

/*&en

def toAsirForm(V) {

@c sort-sm1.push_int0

extern ToAsirForm_V; /* for debug */

@node sm1.push_int0,,, SM1 Functions

if (type(V) == 4) { /* list */

@subsection @code{sm1.push_int0}

if((length(V) == 3) && (V[0] == "sm1_dp")) {

@findex sm1.push_int0

/* For debugging. */

if (ToAsir_Debug != 0) {

ToAsirForm_V = V;

print(map(type,V[1]));

print(V);

}

/* */

Vlist = map(strtov,V[1]);

return(dp_dtop(V[2],Vlist));

} else {

return(map(toAsirForm,V));

}

}else{

return(V);

}

def sm1_toOrdered(V) {

if (type(V) == 4) { /* list */

if((length(V) == 3) && (V[0] == "sm1_dp")) {

Vlist = map(strtov,V[1]);

Ans = "";

F = V[2];

while (F != 0) {

G = dp_hm(F);

F = dp_rest(F);

if (dp_hc(G)>0) {

Ans += "+";

}

Ans += rtostr(dp_dtop(G,Vlist));

}

return Ans;

} else {

return(map(sm1_toOrdered,V));

}

}else{

return(V);

}

def sm1_push_poly0_R(A,P,Vlist) {

return(sm1_push_poly0(P,A,Vlist));

}

def sm1_push_poly0(P,A,Vlist) {

if (type(Vlist[0]) == 4) {

Vlist = Vlist[2];

}

/* if Vlist=[[e,x,y,H,E,Dx,Dy,h],[e,x,y,hH,eE,dx,dy,h],[e,x,y,hH,eE,dx,dy,h]]

list of str (sm1) list of str (asir) list of var (asir)

then we execute the code above.

if (type(A) == 2 || type(A) == 1) { /* recursive poly or number*/

A = dp_ptod(A,Vlist);

ox_push_cmo(P,A);

return;

}

if (type(A) == 0) { /* zero */

sm1(P," (0). ");

return;

}

if (type(A) == 4) { /* list */

ox_execute_string(P," [ ");

map(sm1_push_poly0_R,A,P,Vlist);

ox_execute_string(P," ] ");

return;

}

ox_push_cmo(P,A);

ox_check_errors2(P);

return;

}

/* sm1_push_poly0(0,[0,1,x+y,["Hello",y^3]],[x,y]); */

def sm1_pop_poly0(P,Vlist) {

if (type(Vlist[0]) == 4) {

Vlist = Vlist[2];

}

A = ox_pop_cmo(P);

return(sm1_pop_poly0_0(P,A,Vlist));

}

def sm1_pop_poly0_0_R(A,P,Vlist) {

return(sm1_pop_poly0_0(P,A,Vlist));

}

def sm1_pop_poly0_0(P,A,Vlist) {

if (type(A) == 4) {

return(map(sm1_pop_poly0_0_R,A,P,Vlist));

}

if (type(A)== 9) {return(dp_dtop(A,Vlist));}

return(A);

}

def sm1_push_int0_R(A,P) {

return(sm1_push_int0(P,A));

}

/*&eg-texi

@c sort-sm1_push_int0

@node sm1_push_int0,,, SM1 Functions

@subsection @code{sm1_push_int0}

@findex sm1_push_int0

@table @t

@item sm1_push_int0(@var{p},@var{f})

@item sm1.push_int0(@var{p},@var{f})

:: push the object @var{f} to the server with the descriptor number @var{p}.

@end table

Line 564 Note that @code{ox_push_cmo(@var{p},1234)} send the bi

Line 426 Note that @code{ox_push_cmo(@var{p},1234)} send the bi

@item In other cases, @code{ox_push_cmo} is called without data conversion.

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_push_int0

@c sort-sm1.push_int0

@node sm1_push_int0,,, SM1 Functions

@node sm1.push_int0,,, SM1 Functions

@subsection @code{sm1_push_int0}

@subsection @code{sm1.push_int0}

@findex sm1_push_int0

@findex sm1.push_int0

@table @t

@item sm1_push_int0(@var{p},@var{f})

@item sm1.push_int0(@var{p},@var{f})

:: $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.

@end table

Line 601 Note that @code{ox_push_cmo(@var{p},1234)} send the bi

Line 463 Note that @code{ox_push_cmo(@var{p},1234)} send the bi

/*&C

@example

[219] P=sm1_start();

[219] P=sm1.start();

[220] sm1_push_int0(P,x*dx+1);

[220] sm1.push_int0(P,x*dx+1);

[221] A=ox_pop_cmo(P);

x*dx+1

Line 612 x*dx+1

Line 474 x*dx+1

@end example

@example

[271] sm1_push_int0(0,[x*(x-1),[x]]);

[271] sm1.push_int0(0,[x*(x-1),[x]]);

[272] ox_execute_string(0," deRham ");

Line 620 x*dx+1

Line 482 x*dx+1

[1,2]

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{ox_push_cmo}

@end table

/*&jp-texi

/*&ja

@table @t

@item Reference

@code{ox_push_cmo}

Line 634 x*dx+1

Line 496 x*dx+1

def sm1_push_int0(P,A) {

if (type(A) == 1 || type(A) == 0) {

/* recursive poly or number or 0*/

A = rtostr(A);

ox_push_cmo(P,A);

sm1(P," . (integer) dc ");

return;

}

if (type(A) == 2) {

A = rtostr(A); ox_push_cmo(P,A);

return;

}

if (type(A) == 4) { /* list */

ox_execute_string(P," [ ");

map(sm1_push_int0_R,A,P);

ox_execute_string(P," ] ");

return;

}

ox_push_cmo(P,A);

return;

}

def sm1_push_0_R(A,P) {

/*&en

return(sm1_push_0(P,A));

@c sort-sm1.gb

}

@node sm1.gb,,, SM1 Functions

def sm1_push_0(P,A) {

@subsection @code{sm1.gb}

if (type(A) == 0) {

@findex sm1.gb

/* 0 */

@findex sm1.gb_d

A = rtostr(A);

ox_push_cmo(P,A);

sm1(P," .. ");

return;

}

if (type(A) == 2) {

/* Vlist = vars(A); One should check Vlist is a subset of Vlist3. */

Vlist2 = sm1_vlist(P);

Vlist3 = map(strtov,Vlist2[1]);

B = dp_ptod(A,Vlist3);

ox_push_cmo(P,B);

return;

}

if (type(A) == 4) { /* list */

ox_execute_string(P," [ ");

map(sm1_push_0_R,A,P);

ox_execute_string(P," ] ");

return;

}

ox_push_cmo(P,A);

return;

}

def sm1_push(P,A) {

sm1_push_0(P,A);

}

def sm1_pop(P) {

extern V_sm1_pop;

sm1(P," toAsirForm ");

V_sm1_pop = ox_pop_cmo(P);

return(toAsirForm(V_sm1_pop));

}

def sm1_pop2(P) {

extern V_sm1_pop;

sm1(P," toAsirForm ");

V_sm1_pop = ox_pop_cmo(P);

return([toAsirForm(V_sm1_pop),V_sm1_pop]);

}

def sm1_check_arg_gb(A,Fname) {

/* A = [[x^2+y^2-1,x*y],[x,y],[[x,-1,y,-1]]] */

if (type(A) != 4) {

error(Fname+" : argument should be a list.");

}

if (length(A) < 2) {

error(Fname+" : argument should be a list of 2 or 3 elements.");

}

if (type(A[0]) != 4) {

error(Fname+" : example: [[dx^2+dy^2-4,dx*dy-1]<== it should be a list,[x,y]]");

}

if (!sm1_isListOfPoly(A[0])) {

error(Fname+" : example: [[dx^2+dy^2-4,dx*dy-1]<== it should be a list of polynomials or strings,[x,y]]");

}

if (!sm1_isListOfVar(A[1])) {

error(Fname+" : example: [[dx^2+dy^2-4,dx*dy-1],[x,y]<== list of variables or \"x,y\"]");

}

if (length(A) >= 3) {

if (type(A[2]) != 4) {

error(Fname+" : example:[[dx^2+dy^2-4,dx*dy-1],[x,y],[[x,-1,dx,1]]<== a list of weights]");

}

if (type(A[2][0]) != 4) {

error(Fname+" : example:[[dx^2+dy^2-4,dx*dy-1],[x,y],[[x,-1,dx,1],[dy,1]]<== a list of lists of weight]");

}

return(1);

}

def sm1_isListOfPoly(A) {

if (type(A) !=4 ) return(0);

N = length(A);

for (I=0; I<N; I++) {

if (!(type(A[I]) == 0 || type(A[I]) == 1 || type(A[I]) == 2 ||

type(A[I]) == 7 || type(A[I]) == 9)) {

return(0);

}

return(1);

}

def sm1_isListOfVar(A) {

if (type(A) == 7) return(1); /* "x,y" */

if (type(A) != 4) return(0);

N = length(A);

for (I=0; I<N; I++) {

if (!(type(A[I]) == 2 || type(A[I]) == 7 )) {

return(0);

}

return(1);

}

/*&eg-texi

@c sort-sm1_gb

@node sm1_gb,,, SM1 Functions

@node sm1_gb_d,,, SM1 Functions

@subsection @code{sm1_gb}

@findex sm1_gb

@findex sm1_gb_d

@table @t

@item sm1_gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})

@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})

:: computes the Grobner basis of @var{f} in the ring of differential

operators with the variable @var{v}.

@item sm1_gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})

@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})

:: computes the Grobner basis of @var{f} in the ring of differential

operators with the variable @var{v}.

The result will be returned as a list of distributed polynomials.

Line 791 List

Line 531 List

If @var{w} is not given,

the graded reverse lexicographic order will be used to compute Grobner basis.

@item

The return value of @code{sm1_gb}

The return value of @code{sm1.gb}

is the list of the Grobner basis of @var{f} and the initial

terms (when @var{w} is not given) or initial ideal (when @var{w} is given).

@item

@code{sm1_gb_d} returns the results by a list of distributed polynomials.

@code{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,

Line 805 List

Line 545 List

the homogenized Weyl algebra (See Section 1.2 of the book of SST).

The homogenization variable h is automatically added.

@item

When the optional variable @var{q} is set, @code{sm1_gb} returns,

When the optional variable @var{q} is set, @code{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.

Line 814 List

Line 554 List

the polynomials are dehomogenized (,i.e., h is set to 1).

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_gb

@c sort-sm1.gb

@node sm1_gb,,, SM1 Functions

@node sm1.gb,,, SM1 Functions

@node sm1_gb_d,,, SM1 Functions

@subsection @code{sm1.gb}

@subsection @code{sm1_gb}

@findex sm1.gb

@findex sm1_gb

@findex sm1.gb_d

@findex sm1_gb_d

@table @t

@item sm1_gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})

@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})

:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.

@item sm1_gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})

@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})

:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.

@end table

Line 845 List

Line 584 List

$B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B

$B%V%l%V%J4pDl$r7W;;$9$k(B.

@item

@code{sm1_gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B

@code{sm1.gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B

( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)

$B$N%j%9%H$G$"$k(B.

@item

@code{sm1_gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.

@code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.

$BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.

$BLa$jCM$O(B

[$BJQ?tL>$N%j%9%H(B, $B=g=x$r$-$a$k9TNs(B, $B%0%l%V%J4pDl(B, $B%$%K%7%c%k%b%N%_%"%k$^$?$O%$%K%7%!%kB?9`<0(B]

Line 865 List

Line 604 List

$BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B).

@end itemize

/*&C-texi

/*&C

@example

[293] sm1_gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);

[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]]

@end example

/*&eg-texi

/*&en

In the example above,

@tex the set $\{ x \partial_x + y \partial_y -1,

y^2 \partial_y^2+2\}$

Line 882 The set $\{x \partial_x, y^2 \partial_y\}$ is the lead

Line 621 The set $\{x \partial_x, y^2 \partial_y\}$ is the lead

(the initial monominals) of the Gr\"obner basis.

@end tex

/*&jp-texi

/*&ja

$B>e$NNc$K$*$$$F(B,

@tex $B=89g(B $\{ x \partial_x + y \partial_y -1,

y^2 \partial_y^2+2\}$

Line 894 graded reverse lexicographic order $B$K4X$9$k%0%l%V%J

Line 633 graded reverse lexicographic order $B$K4X$9$k%0%l%V%J

$BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.

@end tex

/*&C-texi

/*&C

@example

[294] sm1_gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);

[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]]

@end example

/*&eg-texi

/*&en

In the example above, two monomials

@tex

$m = x^a y^b \partial_x^c \partial_y^d$ and

Line 913 compared by the reverse lexicographic order

Line 652 compared by the reverse lexicographic order

(i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).

@end tex

/*&jp-texi

/*&ja

$B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B

@tex

$m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B

Line 927 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

Line 666 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

$B$5$l$k(B).

@end tex

/*&C-texi

/*&C

@example

[294] F=sm1_gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);

[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])$

@end example

/*&C-texi

/*&C

@example

[595]

sm1_gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],

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"],

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],

Line 961 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

Line 700 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm1_reduction}, @code{sm1_rat_to_p}

@code{sm1.reduction}, @code{sm1.rat_to_p}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{sm1_reduction}, @code{sm1_rat_to_p}

@code{sm1.reduction}, @code{sm1.rat_to_p}

@end table

def sm1_gb(A) {

SM1_FIND_PROC(P);

P = sm1_check_server(P);

sm1_check_arg_gb(A,"Error in sm1_gb");

sm1_push_int0(P,A);

sm1(P," gb ");

T = sm1_pop2(P);

return(append(T[0],[sm1_toOrdered(T[1])]));

}

def sm1_gb_d(A) {

SM1_FIND_PROC(P);

P = sm1_check_server(P);

sm1_check_arg_gb(A,"Error in sm1_gb_d");

sm1_push_int0(P,A);

sm1(P," gb /gb.tmp1 set ");

sm1(P," gb.tmp1 getOrderMatrix {{(universalNumber) dc} map } map /gb.tmp2 set ");

sm1(P," gb.tmp1 0 get 0 get getvNamesCR { [(class) (indeterminate)] dc } map /gb.tmp3 set ");

sm1(P," gb.tmp1 getRing ring_def "); /* Change the current ring! */

sm1(P,"[[ gb.tmp3 gb.tmp2] gb.tmp1] ");

return(ox_pop_cmo(P));

}

def sm1_pgb(A) {

/*&en

SM1_FIND_PROC(P);

@c sort-sm1.deRham

P = sm1_check_server(P);

@node sm1.deRham,,, SM1 Functions

sm1_check_arg_gb(A,"Error in sm1_pgb");

@subsection @code{sm1.deRham}

sm1(P," set_timer ");

@findex sm1.deRham

sm1_push_int0(P,A);

sm1(P," pgb ");

B = sm1_pop(P);

sm1(P," set_timer ");

return(B);

}

/*&eg-texi

@c sort-sm1_deRham

@node sm1_deRham,,, SM1 Functions

@subsection @code{sm1_deRham}

@findex sm1_deRham

@table @t

@item sm1_deRham([@var{f},@var{v}]|proc=@var{p})

@item sm1.deRham([@var{f},@var{v}]|proc=@var{p})

:: ask the server to evaluate the dimensions of the de Rham cohomology groups

of C^n - (the zero set of @var{f}=0).

@end table

Line 1038 List

Line 744 List

[dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].

@item @var{v} is a list of variables. n = @code{length(@var{v})}.

@item

@code{sm1_deRham} requires huge computer resources.

@code{sm1.deRham} requires huge computer resources.

For example, @code{sm1_deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}

For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}

is already very hard.

@item

To efficiently analyze the roots of b-function, @code{ox_asir} should be used

Line 1048 List

Line 754 List

by the command @*

@code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}

This command is automatically executed when @code{ox_sm1_forAsir} is started.

@item If you make an interruption to the function @code{sm1_deRham}

@item If you make an interruption to the function @code{sm1.deRham}

by @code{ox_reset(Sm1_proc);}, the server might get out of the standard

by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard

mode. So, it is strongly recommended to execute the command

@code{ox_shutdown(Sm1_proc);} to interrupt and restart the server.

@code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_deRham

@c sort-sm1.deRham

@node sm1_deRham,,, SM1 Functions

@node sm1.deRham,,, SM1 Functions

@subsection @code{sm1_deRham}

@subsection @code{sm1.deRham}

@findex sm1_deRham

@findex sm1.deRham

@table @t

@item sm1_deRham([@var{f},@var{v}]|proc=@var{p})

@item sm1.deRham([@var{f},@var{v}]|proc=@var{p})

:: $B6u4V(B C^n - (the zero set of @var{f}=0) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$7$F$/$l$k$h$&$K%5!<%P$KMj$`(B.

@end table

Line 1082 mode. So, it is strongly recommended to execute the co

Line 788 mode. So, it is strongly recommended to execute the co

$B$rLa$9(B.

@item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.

@item

@code{sm1_deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.

@code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.

$B$?$H$($P(B @code{sm1_deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}

$B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}

$B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.

@item

b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}

Line 1092 mode. So, it is strongly recommended to execute the co

Line 798 mode. So, it is strongly recommended to execute the co

$B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.

$B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.

@item

@code{sm1_deRham} $B$r(B @code{ox_reset(Sm1_proc);} $B$GCfCG$9$k$H(B,

@code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,

$B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B

$B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(Sm1_proc);} $B$G(B, @code{ox_sm1_forAsir}

$B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}

$B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[332] sm1_deRham([x^3-y^2,[x,y]]);

[332] sm1.deRham([x^3-y^2,[x,y]]);

[1,1,0]

[333] sm1_deRham([x*(x-1),[x]]);

[333] sm1.deRham([x*(x-1),[x]]);

[1,2]

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm1_start}, @code{deRham} (sm1 command)

@code{sm1.start}, @code{deRham} (sm1 command)

@item 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.

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{sm1_start}, @code{deRham} (sm1 command)

@code{sm1.start}, @code{deRham} (sm1 command)

@item Algorithm:

Oaku, Takayama, An algorithm for de Rham cohomology groups of the

complement of an affine variety via D-module computation,

Line 1128 mode. So, it is strongly recommended to execute the co

Line 834 mode. So, it is strongly recommended to execute the co

def sm1_deRham(A) {

SM1_FIND_PROC(P);

P = sm1_check_server(P);

sm1(P," set_timer ");

sm1_push_int0(P,A);

sm1(P," deRham ");

B = sm1_pop(P);

sm1(P," set_timer ");

ox_check_errors2(P);

return(B);

}

def sm1_vlist(P) {

sm1(P," getvNamesC ");

B=ox_pop_cmo(P);

sm1(P," getvNamesC toAsirVar ");

C=ox_pop_cmo(P);

return([B,C,map(strtov,C)]);

}

/* [ sm1 names(string), asir names(string), asir names(var)] */

/* Vlist = sm1_vlist(P);

sm1_push_poly0( x + 20*x, Vlist[2]);

sm1_pop_poly0(Vlist[2]);

/* ring of Differential operators */

/*&en

def sm1_ringD(V,W) {

@c sort-sm1.hilbert

SM1_FIND_PROC(P);

@node sm1.hilbert,,, SM1 Functions

sm1(P," [ ");

@subsection @code{sm1.hilbert}

if (type(V) == 7) { /* string */

@findex sm1.hilbert

ox_push_cmo(P,V);

}else if (type(V) == 4) {/* list */

V = map(rtostr,V);

ox_push_cmo(P,V);

sm1(P," from_records ");

}else { printf("Error: sm1_ringD"); return(-1); }

sm1(P," ring_of_differential_operators ");

if (type(W) != 0) {

sm1_push_int0(P,W); sm1(P," weight_vector ");

}

sm1(P," pstack ");

sm1(P," 0 ] define_ring getOrderMatrix {{(universalNumber) dc}map}map ");

ox_check_errors2(P);

M = ox_pop_cmo(P);

return([sm1_vlist(P)[2],M]);

}

def sm1_expand_d(F) {

SM1_FIND_PROC(P);

ox_push_cmo(P,F);

sm1(P, " expand ");

return(ox_pop_cmo(P));

}

def sm1_mul_d(A,B) {

SM1_FIND_PROC(P);

ox_push_cmo(P,A);

ox_push_cmo(P,B);

sm1(P," mul ");

return(ox_pop_cmo(P));

}

def sm1_dehomogenize_d(A) {

SM1_FIND_PROC(P);

ox_push_cmo(P,A);

sm1(P," dehomogenize ");

return(ox_pop_cmo(P));

}

def sm1_homogenize_d(A) {

SM1_FIND_PROC(P);

ox_push_cmo(P,A);

sm1(P," homogenize ");

return(ox_pop_cmo(P));

}

def sm1_groebner_d(A) {

SM1_FIND_PROC(P);

ox_push_cmo(P,A);

sm1(P," groebner ");

return(ox_pop_cmo(P));

}

def sm1_reduction_d(F,G) {

SM1_FIND_PROC(P);

ox_push_cmo(P,F);

ox_push_cmo(P,G);

sm1(P," reduction ");

return(ox_pop_cmo(P));

}

def sm1_reduction_noH_d(F,G) {

SM1_FIND_PROC(P);

ox_push_cmo(P,F);

ox_push_cmo(P,G);

sm1(P," reduction-noH ");

return(ox_pop_cmo(P));

}

/*&eg-texi

@c sort-sm1_hilbert

@node sm1_hilbert,,, SM1 Functions

@subsection @code{sm1_hilbert}

@findex sm1_hilbert

@findex hilbert_polynomial

@table @t

@item sm1_hilbert([@var{f},@var{v}]|proc=@var{p})

@item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})

:: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.

@item hilbert_polynomial(@var{f},@var{v})

:: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.

Line 1259 List

Line 867 List

degree is less than or equal to k and I is the ideal generated by the

set of polynomials @var{f}.

@item

Note for sm1_hilbert:

Note for sm1.hilbert:

For an efficient computation, it is preferable that

the set of polynomials @var{f} is a set of monomials.

In fact, this function firstly compute a Grobner basis of @var{f}, and then

Line 1270 List

Line 878 List

polynomials in @code{sm1} is slower than in @code{asir}.

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_hilbert

@c sort-sm1.hilbert

@node sm1_hilbert,,, SM1 Functions

@node sm1.hilbert,,, SM1 Functions

@subsection @code{sm1_hilbert}

@subsection @code{sm1.hilbert}

@findex sm1_hilbert

@findex sm1.hilbert

@findex hilbert_polynomial

@table @t

@item sm1_hilbert([@var{f},@var{v}]|proc=@var{p})

@item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})

:: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.

@item hilbert_polynomial(@var{f},@var{v})

:: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.

Line 1299 List

Line 907 List

h(k) = dim_Q F_k/I \cap F_k $B$3$3$G(B F_k $B$O<!?t$,(B k $B0J2<$G$"$k$h$&$J(B

$BB?9`<0$N=89g$G$"$k(B. I $B$OB?9`<0$N=89g(B @var{f} $B$G@8@.$5$l$k%$%G%"%k$G$"$k(B.

@item

sm1_hilbert $B$K$+$s$9$k%N!<%H(B:

sm1.hilbert $B$K$+$s$9$k%N!<%H(B:

$B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.

$B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial

monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.

Line 1309 List

Line 917 List

@end itemize

/*&C-texi

/*&C

@example

[346] load("katsura")$

Line 1334 List

Line 942 List

[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]]);

[284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm1_start}, @code{sm1_gb}, @code{longname}

@code{sm1.start}, @code{sm1.gb}, @code{longname}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{sm1_start}, @code{sm1_gb}, @code{longname}

@code{sm1.start}, @code{sm1.gb}, @code{longname}

@end table

def sm1_hilbert(A) {

SM1_FIND_PROC(P);

P = sm1_check_server(P);

sm1(P,"[ ");

sm1_push_int0(P,A[0]);

sm1_push_int0(P,A[1]);

sm1(P," ] pgb /sm1_hilbert.gb set ");

sm1(P," sm1_hilbert.gb 0 get { init toString } map ");

sm1_push_int0(P,A[1]);

sm1(P, " hilbert ");

B = sm1_pop(P);

return(B[1]/fac(B[0]));

}

/*&eg-texi

/*&en

@c sort-sm1_genericAnn

@c sort-sm1.genericAnn

@node sm1_genericAnn,,, SM1 Functions

@node sm1.genericAnn,,, SM1 Functions

@subsection @code{sm1_genericAnn}

@subsection @code{sm1.genericAnn}

@findex sm1_genericAnn

@findex sm1.genericAnn

@table @t

@item sm1_genericAnn([@var{f},@var{v}]|proc=@var{p})

@item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})

:: It computes the annihilating ideal for @var{f}^s.

@var{v} is the list of variables. Here, s is @var{v}[0] and

@var{f} is a polynomial in the variables @code{rest}(@var{v}).

Line 1395 List

Line 990 List

@var{f} is a polynomial in the variables @code{rest}(@var{v}).

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_genericAnn

@c sort-sm1.genericAnn

@node sm1_genericAnn,,, SM1 Functions

@node sm1.genericAnn,,, SM1 Functions

@subsection @code{sm1_genericAnn}

@subsection @code{sm1.genericAnn}

@findex sm1_genericAnn

@findex sm1.genericAnn

@table @t

@item sm1_genericAnn([@var{f},@var{v}]|proc=@var{p})

@item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})

:: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.

@var{v} $B$OJQ?t$N%j%9%H$G$"$k(B. $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,

@var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.

Line 1425 List

Line 1020 List

@var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[595] sm1_genericAnn([x^3+y^3+z^3,[s,x,y,z]]);

[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]

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm1_start}

@code{sm1.start}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{sm1_start}

@code{sm1.start}

@end table

def sm1_genericAnn(F) {

SM1_FIND_PROC(P);

sm1_push_int0(P,F[0]);

sm1_push_int0(P,F[1]);

sm1(P, " genericAnn ");

B = sm1_pop(P);

return(B);

}

def sm1_tensor0(F) {

/*&en

SM1_FIND_PROC(P);

@c sort-sm1.wTensor0

sm1_push_int0(P,F);

@node sm1.wTensor0,,, SM1 Functions

sm1(P, " tensor0 ");

@subsection @code{sm1.wTensor0}

B = sm1_pop(P);

@findex sm1.wTensor0

return(B);

}

/*&eg-texi

@c sort-sm1_wTensor0

@node sm1_wTensor0,,, SM1 Functions

@subsection @code{sm1_wTensor0}

@findex sm1_wTensor0

@table @t

@item sm1_wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

@item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

:: It computes the D-module theoretic 0-th tensor product

of @var{f} and @var{g}.

@end table

Line 1491 List

Line 1070 List

@var{w} is a list of weights. The integer @var{w}[i] is

the weight of the variable @var{v}[i].

@item

@code{sm1_wTensor0} calls @code{wRestriction0} of @code{ox_sm1},

@code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},

which requires a generic weight

vector @var{w} to compute the restriction.

If @var{w} is not generic, the computation fails.

Line 1503 the inputs @var{f} and @var{g} are left ideals of D.

Line 1082 the inputs @var{f} and @var{g} are left ideals of D.

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_wTensor0

@c sort-sm1.wTensor0

@node sm1_wTensor0,,, SM1 Functions

@node sm1.wTensor0,,, SM1 Functions

@subsection @code{sm1_wTensor0}

@subsection @code{sm1.wTensor0}

@findex sm1_wTensor0

@findex sm1.wTensor0

@table @t

@item sm1_wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

@item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

:: @var{f} $B$H(B @var{g} $B$N(B D-module $B$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r(B

$B7W;;$9$k(B.

@end table

Line 1532 the inputs @var{f} and @var{g} are left ideals of D.

Line 1111 the inputs @var{f} and @var{g} are left ideals of D.

@var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.

$B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.

@item

@code{sm1_wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}

@code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}

$B$r$h$s$G$$$k(B.

@code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}

$B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.

Line 1543 the inputs @var{f} and @var{g} are left ideals of D.

Line 1122 the inputs @var{f} and @var{g} are left ideals of D.

$B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[258] sm1_wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);

[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]]

Line 1553 the inputs @var{f} and @var{g} are left ideals of D.

Line 1132 the inputs @var{f} and @var{g} are left ideals of D.

def sm1_wTensor0(F) {

SM1_FIND_PROC(P);

sm1_push_int0(P,F);

sm1(P, " wTensor0 ");

B = sm1_pop(P);

return(B);

}

/*&en

/*&eg-texi

@c sort-sm1.reduction

@c sort-sm1_reduction

@node sm1.reduction,,, SM1 Functions

@node sm1_reduction,,, SM1 Functions

@subsection @code{sm1.reduction}

@subsection @code{sm1_reduction}

@findex sm1.reduction

@findex sm1_reduction

@table @t

@item sm1_reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

@end table

Line 1588 Number (the process number of ox_sm1)

Line 1159 Number (the process number of ox_sm1)

in the homogenized Weyl algebra; it applies the

division algorithm to @var{f}. The set of variables is @var{v} and

@var{w} is weight vectors to determine the order, which can be ommited.

@code{sm1_reduction_noH} is for the Weyl algebra.

@code{sm1.reduction_noH} is for the Weyl algebra.

@item The return value is of the form

[r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and

r/c0 + c1 g1 + ... + cm gm = 0.

c0 f + c1 g1 + ... + cm gm = r.

r/c0 is the normal form.

@item The function reduction reduces reducible terms that appear

in lower order terms.

@item The functions

sm1_reduction_d(P,F,G) and sm1_reduction_noH_d(P,F,G)

sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)

are for distributed polynomials.

@end itemize

/*&jp-texi

/*&ja

@node sm1_reduction,,, SM1 Functions

@node sm1.reduction,,, SM1 Functions

@subsection @code{sm1_reduction}

@subsection @code{sm1.reduction}

@findex sm1_reduction

@findex sm1.reduction

@table @t

@item sm1_reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})

@end table

Line 1627 are for distributed polynomials.

Line 1198 are for distributed polynomials.

$BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.

@var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,

$B>JN,$7$F$b$h$$(B.

@code{sm1_reduction_noH} $B$O(B, Weyl algebra $BMQ(B.

@code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.

@item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:

[r,c0,[c1,...,cm],[g1,...gm]] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,

[r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,

r/c0 + c1 g1 + ... + cm gm = 0

c0 f + c1 g1 + ... + cm gm = r

$B$,$J$j$?$D(B.

r/c0 $B$,(B normal form $B$G$"$k(B.

@item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.

@item $BH!?t(B

sm1_reduction_d(P,F,G) $B$*$h$S(B sm1_reduction_noH_d(P,F,G)

sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)

$B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[259] sm1_reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);

[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],[x+y^3-4*y,y^4-4*y^2+1]]

[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]]]);

[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],[x+y^3-4*y,y^4-4*y^2+1]]

[0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm1_start}, @code{sm1_find_proc}, @code{d_true_nf}

@code{sm1.start}, @code{d_true_nf}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{sm1_start}, @code{sm1_find_proc}, @code{d_true_nf}

@code{sm1.start}, @code{d_true_nf}

@end table

def sm1_reduction(A) {

/* Example: sm1_reduction(A|proc=10) */

SM1_FIND_PROC(P);

/* check the arguments */

if (type(A) != 4) {

error("sm1_reduction(A|proc=p): A must be a list.");

}

AA = [rtostr(A[0])];

AA = append(AA,[ map(rtostr,A[1]) ]);

AA = append(AA, cdr(cdr(A)));

sm1(P," /reduction*.noH 0 def ");

sm1_push_int0(P,AA);

sm1(P," reduction* ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_reduction_noH(A) {

/*&en

/* Example: sm1_reduction(A|proc=10) */

@node sm1.xml_tree_to_prefix_string,,, SM1 Functions

SM1_FIND_PROC(P);

@subsection @code{sm1.xml_tree_to_prefix_string}

/* check the arguments */

@findex sm1.xml_tree_to_prefix_string

if (type(A) != 4) {

error("sm1_reduction_noH(A|proc=p): A must be a list.");

}

AA = [rtostr(A[0])];

AA = append(AA,[ map(rtostr,A[1]) ]);

AA = append(AA, cdr(cdr(A)));

sm1(P," /reduction*.noH 1 def ");

sm1_push_int0(P,AA);

sm1(P," reduction* ");

ox_check_errors2(P);

return(sm1_pop(P));

}

/*&eg-texi

@node sm1_xml_tree_to_prefix_string,,, SM1 Functions

@subsection @code{sm1_xml_tree_to_prefix_string}

@findex sm1_xml_tree_to_prefix_string

@table @t

@item sm1_xml_tree_to_prefix_string(@var{s}|proc=@var{p})

@item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})

:: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.

@end table

Line 1722 asir has not yet understood this CMO.

Line 1260 asir has not yet understood this CMO.

command search path.)

@end itemize

/*&jp-texi

/*&ja

@node sm1_xml_tree_to_prefix_string,,, SM1 Functions

@node sm1.xml_tree_to_prefix_string,,, SM1 Functions

@subsection @code{sm1_xml_tree_to_prefix_string}

@subsection @code{sm1.xml_tree_to_prefix_string}

@findex sm1_xml_tree_to_prefix_string

@findex sm1.xml_tree_to_prefix_string

@table @t

@item sm1_xml_tree_to_prefix_string(@var{s}|proc=@var{p})

@item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})

:: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.

@end table

Line 1749 String

Line 1287 String

($B$?$H$($P(B, /usr/local/jdk1.1.8/bin $B$r%3%^%s%I%5!<%A%Q%9$KF~$l$k$J$I(B.)

@end itemize

/*&C-texi

/*&C

@example

[263] load("om");

Line 1762 Trying to connect to the server... Done.

Line 1300 Trying to connect to the server... Done.

[271] sm1_xml_tree_to_prefix_string(F);

[271] sm1.xml_tree_to_prefix_string(F);

basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}

Line 1780 basic_plus(basic_times(basic_power(x,4),1),basic_times

Line 1318 basic_plus(basic_times(basic_power(x,4),1),basic_times

def sm1_xml_tree_to_prefix_string(A) {

SM1_FIND_PROC(P);

/* check the arguments */

if (type(A) != 7) {

error("sm1_xml_tree_to_prefix_string(A|proc=p): A must be a string.");

}

ox_push_cmo(P,A);

sm1(P," xml_tree_to_prefix_string ");

ox_check_errors2(P);

return(ox_pop_cmo(P));

}

def sm1_wbf(A) {

/*&en

SM1_FIND_PROC(P);

@c sort-sm1.syz

/* check the arguments */

@node sm1.syz,,, SM1 Functions

if (type(A) != 4) {

@subsection @code{sm1.syz}

error("sm1_wbf(A): A must be a list.");

@findex sm1.syz

}

@findex sm1.syz_d

if (length(A) != 3) {

error("sm1_wbf(A): A must be a list of the length 3.");

}

if (type(A[0]) != 4 || type(A[1]) != 4 || type(A[2]) != 4) {

error("sm1_wbf([A,B,C]): A, B, C must be a list.");

}

if (! (type(A[2][0]) == 7 || type(A[2][0]) == 2)) {

error("sm1_wbf([A,B,C]): C must be of a form [v-name, v-weight, ...]");

}

sm1_push_int0(P,A);

sm1(P," wbf ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_wbfRoots(A) {

SM1_FIND_PROC(P);

/* check the arguments */

if (type(A) != 4) {

error("sm1_wbfRoots(A): A must be a list.");

}

if (length(A) != 3) {

error("sm1_wbfRoots(A): A must be a list of the length 3.");

}

if (type(A[0]) != 4 || type(A[1]) != 4 || type(A[2]) != 4) {

error("sm1_wbfRoots([A,B,C]): A, B, C must be a list.");

}

if (! (type(A[2][0]) == 7 || type(A[2][0]) == 2)) {

error("sm1_wbfRoots([A,B,C]): C must be of a form [v-name, v-weight, ...]");

}

sm1_push_int0(P,A);

sm1(P," wbfRoots ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_res_div(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,[[A[0],A[1]],A[2]]);

sm1(P," res*div ");

ox_check_errors2(P);

return(sm1_pop(P));

}

/*&eg-texi

@c sort-sm1_syz

@node sm1_syz,,, SM1 Functions

@node sm1_syz_d,,, SM1 Functions

@subsection @code{sm1_syz}

@findex sm1_syz

@findex sm1_syz_d

@table @t

@item sm1_syz([@var{f},@var{v},@var{w}]|proc=@var{p})

@item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})

:: computes the syzygy of @var{f} in the ring of differential

operators with the variable @var{v}.

@end table

Line 1874 Here @var{s} is the syzygy of @var{f} in the ring of d

Line 1349 Here @var{s} is the syzygy of @var{f} in the ring of d

operators with the variable @var{v}.

@var{g} is a Groebner basis of @var{f} with the weight vector @var{w},

and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner

basis @var {g}.

basis @var{g}.

@var{t} is the syzygy of the Gr\"obner basis @var{g}.

In summary, @var{g} = @var{m} @var{f} and

@var{s} @var{f} = 0 hold as matrices.

Line 1888 In summary, @var{g} = @var{m} @var{f} and

Line 1363 In summary, @var{g} = @var{m} @var{f} and

The homogenization variable h is automatically added.

@end itemize

/*&jp-texi

/*&ja

@c sort-sm1_syz

@c sort-sm1.syz

@node sm1_syz,,, SM1 Functions

@node sm1.syz,,, SM1 Functions

@node sm1_syz_d,,, SM1 Functions

@subsection @code{sm1.syz}

@subsection @code{sm1_syz}

@findex sm1.syz

@findex sm1_syz

@findex sm1.syz_d

@findex sm1_syz_d

@table @t

@item sm1_syz([@var{f},@var{v},@var{w}]|proc=@var{p})

@item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})

:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.

@end table

Line 1931 syzygy $B$G$"$k(B.

Line 1405 syzygy $B$G$"$k(B.

$BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.

@end itemize

/*&C-texi

/*&C

@example

[293] sm1_syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);

[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]]]]

@end example

/*&C-texi

/*&C

@example

[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]]]);

[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]],

Line 1952 syzygy $B$G$"$k(B.

Line 1426 syzygy $B$G$"$k(B.

def sm1_syz(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,A);

sm1(P," syz ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_res_solv(A) {

/*&en

SM1_FIND_PROC(P);

@node sm1.mul,,, SM1 Functions

sm1_push_int0(P,[[A[0],A[1]],A[2]]);

@subsection @code{sm1.mul}

sm1(P," res*solv ");

@findex sm1.mul

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_res_solv_h(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,[[A[0],A[1]],A[2]]);

sm1(P," res*solv*h ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_mul(A,B,V) {

SM1_FIND_PROC(P);

sm1_push_int0(P,[[A,B],V]);

sm1(P," res*mul ");

ox_check_errors2(P);

return(sm1_pop(P));

}

/*&eg-texi

@node sm1_mul,,, SM1 Functions

@subsection @code{sm1_mul}

@findex sm1_mul

@table @t

@item sm1_mul(@var{f},@var{g},@var{v}|proc=@var{p})

@item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})

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

@end table

Line 2007 List

Line 1449 List

@itemize @bullet

@item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.

@item @code{sm1_mul_h} is for homogenized Weyl algebra.

@item @code{sm1.mul_h} is for homogenized Weyl algebra.

@end itemize

/*&jp-texi

/*&ja

@node sm1_mul,,, SM1 Functions

@node sm1.mul,,, SM1 Functions

@subsection @code{sm1_mul}

@subsection @code{sm1.mul}

@findex sm1_mul

@findex sm1.mul

@table @t

@item sm1_mul(@var{f},@var{g},@var{v}|proc=@var{p})

@item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})

:: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}

$B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.

@end table

Line 2035 List

Line 1477 List

@itemize @bullet

@item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}

$B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.

@item @code{sm1_mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.

@item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.

@end itemize

/*&C-texi

/*&C

@example

[277] sm1_mul(dx,x,[x]);

[277] sm1.mul(dx,x,[x]);

x*dx+1

[278] sm1_mul([x,y],[1,2],[x,y]);

[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]);

[279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);

[[x+2,y+4],[3*x+4,3*y+8]]

@end example

Line 2054 x+2*y

Line 1496 x+2*y

def sm1_mul_h(A,B,V) {

SM1_FIND_PROC(P);

sm1_push_int0(P,[[A,B],V]);

sm1(P," res*mul*h ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_adjoint(A,V) {

/*&en

SM1_FIND_PROC(P);

@node sm1.distraction,,, SM1 Functions

sm1_push_int0(P,[A,V]);

@subsection @code{sm1.distraction}

sm1(P," res*adjoint ");

@findex sm1.distraction

ox_check_errors2(P);

return(sm1_pop(P));

}

def transpose(A) {

if (type(A) == 4) {

N = length(A); M = length(A[0]);

B = newmat(N,M,A);

C = newmat(M,N);

for (I=0; I<N; I++) {

for (J=0; J<M; J++) {

C[J][I] = B[I][J];

}

D = newvect(M);

for (J=0; J<M; J++) {

D[J] = C[J];

}

return(map(vtol,vtol(D)));

}else{

print(A)$

error("tranpose: traspose for this argument has not been implemented.");

}

def sm1_resol1(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,A);

sm1(P," res*resol1 ");

ox_check_errors2(P);

return(sm1_pop(P));

}

def sm1_gcd_aux(A,B) {

if (type(A) == 1 && type(B) == 1) return(igcd(A,B));

else return(gcd(A,B));

}

def sm1_lcm_aux(V) { /* sm1_lcm_aux([3,5,6]); */

N = length(V);

if (N == 0) return(0);

if (N == 1) return(V[0]);

L = V[0];

for (I=1; I<N; I++) {

L = red(L*V[I]/sm1_gcd_aux(L,V[I]));

}

return(L);

}

def sm1_mul_v(V,S) {

if (type(V) == 4) {

return(map(sm1_mul_v,V,S));

} else {

return(V*S);

}

def sm1_div_v(V,S) {

if (type(V) == 4) {

return(map(sm1_div_v,V,S));

} else {

return(V/S);

}

def sm1_rat_to_p_aux(T) { /* cf. sm1_rat2plist2 */

T = red(T);

T1 = nm(T); T1a = ptozp(T1);

T1b = red(T1a/T1);

T2 = dn(T);

return([T1a*dn(T1b),T2*nm(T1b)]);

}

def sm1_denom_aux0(A) {

return(A[1]);

}

def sm1_num_aux0(P) {

return(P[0]);

}

def sm1_rat_to_p(T) {

if (type(T) == 4) {

A = map(sm1_rat_to_p,T);

D = map(sm1_denom_aux0,A);

N = map(sm1_num_aux0,A);

L = sm1_lcm_aux(D);

B = newvect(length(N));

for (I=0; I<length(N); I++) {

B[I] = sm1_mul_v(N[I],L/D[I]);

}

return([vtol(B),L]);

}else{

return(sm1_rat_to_p_aux(T));

}

/* ---------------------------------------------- */

def sm1_distraction(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,A);

sm1(P," distraction2* ");

ox_check_errors2(P);

return(sm1_pop(P));

}

/*&eg-texi

@node sm1_distraction,,, SM1 Functions

@subsection @code{sm1_distraction}

@findex sm1_distraction

@table @t

@item sm1_distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})

@item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})

:: ask the @code{sm1} server to compute the distraction of @var{f}.

@end table

Line 2206 See Saito, Sturmfels, Takayama : Grobner Deformations

Line 1528 See Saito, Sturmfels, Takayama : Grobner Deformations

@end itemize

/*&jp-texi

/*&ja

@node sm1_distraction,,, SM1 Functions

@node sm1.distraction,,, SM1 Functions

@subsection @code{sm1_distraction}

@subsection @code{sm1.distraction}

@findex sm1_distraction

@findex sm1.distraction

@table @t

@item sm1_distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})

@item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})

:: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.

@end table

Line 2237 See Saito, Sturmfels, Takayama : Grobner Deformations

Line 1559 See Saito, Sturmfels, Takayama : Grobner Deformations

@end itemize

/*&C-texi

/*&C

@example

[280] sm1_distraction([x*dx,[x],[x],[dx],[x]]);

[280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);

[281] sm1_distraction([dx^2,[x],[x],[dx],[x]]);

[281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);

x^2-x

[282] sm1_distraction([x^2,[x],[x],[dx],[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]]);

[284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);

(x^2-x)*dy+x*y

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{distraction2(sm1)},

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{distraction2(sm1)},

@end table

/* Temporary functions */

/* Use this function for a while to wait a fix of asir. */

def sm1_ntoint32(I) { /* Fixed */

SM1_FIND_PROC(P);

if (I >= 0) return(ntoint32(I));

sm1(P," "+rtostr(I)+" ");

return(ox_pop_cmo(P));

}

def sm1_to_ascii_array(S) { /* Use strtoascii */

SM1_FIND_PROC(P);

ox_push_cmo(P,S);

sm1(P," (array) dc { (universalNumber) dc } map ");

return(ox_pop_cmo(P));

}

def sm1_from_ascii_array(S) { /* Use asciitostr */

SM1_FIND_PROC(P);

ox_push_cmo(P,S);

sm1(P," { (integer) dc (string) dc } map cat ");

return(ox_pop_cmo(P));

}

[288] sm1_to_ascii_array("Hello");

[72,101,108,108,111]

[289] sm1_from_ascii_array(@@);

Hello

/* end of temporary functions */

/*&en

@node sm1.gkz,,, SM1 Functions

def sm1_gkz(S) {

@subsection @code{sm1.gkz}

SM1_FIND_PROC(P);

@findex sm1.gkz

A = S[0];

B = S[1];

AA = [ ];

BB = [ ];

for (I=0; I<length(A); I++) {

AA = append(AA,[map(ntoint32,A[I])]);

BB = append(BB,[ntoint32(0)]);

}

sm1(P,"[ ");

sm1_push_int0(P,AA);

sm1_push_int0(P,BB);

sm1(P," ] gkz ");

ox_check_errors2(P);

R = sm1_pop(P);

RR0 = map(eval_str,R[0]);

RR1 = map(eval_str,R[1]);

RR3 = [ ];

for (I=0; I<length(B); I++) {

RR3 = append(RR3,[ sm1_rat_to_p(RR0[I]-B[I])[0] ]);

}

for (I=length(B); I<length(RR0); I++) {

RR3 = append(RR3,[RR0[I]]);

}

return([RR3,RR1]);

}

/*&eg-texi

@node sm1_gkz,,, SM1 Functions

@subsection @code{sm1_gkz}

@findex sm1_gkz

@table @t

@item sm1_gkz([@var{A},@var{B}]|proc=@var{p})

@item sm1.gkz([@var{A},@var{B}]|proc=@var{p})

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

@var{A} with the parameter vector @var{B}.

@end table

Line 2351 List

Line 1616 List

@end itemize

/*&jp-texi

/*&ja

@node sm1_gkz,,, SM1 Functions

@node sm1.gkz,,, SM1 Functions

@subsection @code{sm1_gkz}

@subsection @code{sm1.gkz}

@findex sm1_gkz

@findex sm1.gkz

@table @t

@item sm1_gkz([@var{A},@var{B}]|proc=@var{p})

@item sm1.gkz([@var{A},@var{B}]|proc=@var{p})

:: $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.

@end table

Line 2374 List

Line 1639 List

@end itemize

/*&C-texi

/*&C

@example

[280] sm1_gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);

[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]]

Line 2388 List

Line 1653 List

def sm1_appell1(S) {

N = length(S)-2;

B = cdr(cdr(S));

A = S[0];

C = S[1];

V = [ ];

for (I=0; I<N; I++) {

V = append(V,[sm1aux_x(I+1)]);

}

Ans = [ ];

Euler = 0;

for (I=0; I<N; I++) {

Euler = sm1aux_x(I+1)*sm1aux_dx(I+1) + Euler;

}

for (I=0; I<N; I++) {

T = sm1_mul(sm1aux_dx(I+1), Euler+C-1,V)-

sm1_mul(Euler+A, sm1aux_x(I+1)*sm1aux_dx(I+1)+B[I],V);

/* Tmp=sm1_rat_to_p(T);

print(Tmp[0]/Tmp[1]-T)$ */

T = sm1_rat_to_p(T)[0];

Ans = append(Ans,[T]);

}

for (I=0; I<N; I++) {

for (J=I+1; J<N; J++) {

T = (sm1aux_x(I+1)-sm1aux_x(J+1))*sm1aux_dx(I+1)*sm1aux_dx(J+1)

- B[J]*sm1aux_dx(I+1) + B[I]*sm1aux_dx(J+1);

/* Tmp=sm1_rat_to_p(T);

print(Tmp[0]/Tmp[1]-T)$ */

T = sm1_rat_to_p(T)[0];

Ans = append(Ans,[T]);

}

return([Ans,V]);

}

def sm1aux_dx(I) {

/*&en

return(strtov("dx"+rtostr(I)));

@node sm1.appell1,,, SM1 Functions

}

@subsection @code{sm1.appell1}

def sm1aux_x(I) {

@findex sm1.appell1

return(strtov("x"+rtostr(I)));

}

/*&eg-texi

@node sm1_appell1,,, SM1 Functions

@subsection @code{sm1_appell1}

@findex sm1_appell1

@table @t

@item sm1_appell1(@var{a}|proc=@var{p})

@item sm1.appell1(@var{a}|proc=@var{p})

:: Returns the Appell hypergeometric system F_1 or F_D.

@end table

Line 2460 The parameters a, c, b1, ..., bn may be rational numbe

Line 1682 The parameters a, c, b1, ..., bn may be rational numbe

@end itemize

/*&jp-texi

/*&ja

@node sm1_appell1,,, SM1 Functions

@node sm1.appell1,,, SM1 Functions

@subsection @code{sm1_appell1}

@subsection @code{sm1.appell1}

@findex sm1_appell1

@findex sm1.appell1

@table @t

@item sm1_appell1(@var{a}|proc=@var{p})

@item sm1.appell1(@var{a}|proc=@var{p})

:: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.

@end table

Line 2487 F_D(a,b1,b2,...,bn,c;x1,...,xn)

Line 1709 F_D(a,b1,b2,...,bn,c;x1,...,xn)

@end itemize

/*&C-texi

/*&C

@example

[281] sm1_appell1([1,2,3,4]);

[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(@@);

[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,

Line 2505 F_D(a,b1,b2,...,bn,c;x1,...,xn)

Line 1727 F_D(a,b1,b2,...,bn,c;x1,...,xn)

+(-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]));

[283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));

[285] Mu=2$ Beta = 1/3$

[287] sm1_rank(sm1_appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));

[287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));

Line 2517 F_D(a,b1,b2,...,bn,c;x1,...,xn)

Line 1739 F_D(a,b1,b2,...,bn,c;x1,...,xn)

def sm1_appell4(S) {

/*&en

N = length(S)-2;

@node sm1.appell4,,, SM1 Functions

B = cdr(cdr(S));

@subsection @code{sm1.appell4}

A = S[0];

@findex sm1.appell4

C = S[1];

V = [ ];

for (I=0; I<N; I++) {

V = append(V,[sm1aux_x(I+1)]);

}

Ans = [ ];

Euler = 0;

for (I=0; I<N; I++) {

Euler = sm1aux_x(I+1)*sm1aux_dx(I+1) + Euler;

}

for (I=0; I<N; I++) {

T = sm1_mul(sm1aux_dx(I+1), sm1aux_x(I+1)*sm1aux_dx(I+1)+B[I]-1,V)-

sm1_mul(Euler+A,Euler+C,V);

/* Tmp=sm1_rat_to_p(T);

print(Tmp[0]/Tmp[1]-T)$ */

T = sm1_rat_to_p(T)[0];

Ans = append(Ans,[T]);

}

return([Ans,V]);

}

/*&eg-texi

@node sm1_appell4,,, SM1 Functions

@subsection @code{sm1_appell4}

@findex sm1_appell4

@table @t

@item sm1_appell4(@var{a}|proc=@var{p})

@item sm1.appell4(@var{a}|proc=@var{p})

:: Returns the Appell hypergeometric system F_4 or F_C.

@end table

Line 2569 The parameters a, b, c1, ..., cn may be rational numbe

Line 1766 The parameters a, b, c1, ..., cn may be rational numbe

@end itemize

/*&jp-texi

/*&ja

@node sm1_appell4,,, SM1 Functions

@node sm1.appell4,,, SM1 Functions

@subsection @code{sm1_appell4}

@subsection @code{sm1.appell4}

@findex sm1_appell4

@findex sm1.appell4

@table @t

@item sm1_appell4(@var{a}|proc=@var{p})

@item sm1.appell4(@var{a}|proc=@var{p})

:: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.

@end table

Line 2596 F_C(a,b,c1,c2,...,cn;x1,...,xn)

Line 1793 F_C(a,b,c1,c2,...,cn;x1,...,xn)

@end itemize

/*&C-texi

/*&C

@example

[281] sm1_appell4([1,2,3,4]);

[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(@@);

[282] sm1.rank(@@);

@end example

Line 2614 F_C(a,b,c1,c2,...,cn;x1,...,xn)

Line 1811 F_C(a,b,c1,c2,...,cn;x1,...,xn)

def sm1_rank(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,A);

sm1(P," rank toString .. ");

ox_check_errors2(P);

R = sm1_pop(P);

return(R);

}

def sm1_rrank(A) {

SM1_FIND_PROC(P);

sm1_push_int0(P,A);

sm1(P," rrank toString .. ");

ox_check_errors2(P);

R = sm1_pop(P);

return(R);

}

/*&en

/*&eg-texi

@node sm1.rank,,, SM1 Functions

@node sm1_rank,,, SM1 Functions

@subsection @code{sm1.rank}

@subsection @code{sm1_rank}

@findex sm1.rank

@findex sm1_rank

@table @t

@item sm1_rank(@var{a}|proc=@var{p})

@item sm1.rank(@var{a}|proc=@var{p})

:: Returns the holonomic rank of the system of differential equations @var{a}.

@end table

Line 2657 at a generic point of the system of differential equat

Line 1837 at a generic point of the system of differential equat

The dimension is called the holonomic rank.

@item @var{a} is a list consisting of a list of differential equations and

a list of variables.

@item @code{sm1_rrank} returns the holonomic rank when @var{a} is regular

@item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular

holonomic. It is generally faster than @code{sm1_rank}.

holonomic. It is generally faster than @code{sm1.rank}.

@end itemize

/*&jp-texi

/*&ja

@node sm1_rank,,, SM1 Functions

@node sm1.rank,,, SM1 Functions

@subsection @code{sm1_rank}

@subsection @code{sm1.rank}

@findex sm1_rank

@findex sm1.rank

@table @t

@item sm1_rank(@var{a}|proc=@var{p})

@item sm1.rank(@var{a}|proc=@var{p})

:: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.

@end table

Line 2684 holonomic. It is generally faster than @code{sm1_rank}

Line 1864 holonomic. It is generally faster than @code{sm1_rank}

@item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B

$BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.

@item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.

@item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1_rrank}

@item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}

$B$b(B holonomic rank $B$rLa$9(B.

$B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1_rank} $B$h$jAa$$(B.

$B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.

@end itemize

/*&C-texi

/*&C

@example

[284] sm1_gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);

[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(@@);

[285] sm1.rrank(@@);

[286] sm1_gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);

[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(@@);

[287] sm1.rrank(@@);

@end example

def sm1_auto_reduce(T) {

SM1_FIND_PROC(P);

sm1(P,"[(AutoReduce) "+rtostr(T)+" ] system_variable ");

ox_check_errors2(P);

R = sm1_pop(P);

return(R);

}

/*&eg-texi

/*&en

@node sm1_auto_reduce,,, SM1 Functions

@node sm1.auto_reduce,,, SM1 Functions

@subsection @code{sm1_auto_reduce}

@subsection @code{sm1.auto_reduce}

@findex sm1_auto_reduce

@findex sm1.auto_reduce

@table @t

@item sm1_auto_reduce(@var{s}|proc=@var{p})

@item sm1.auto_reduce(@var{s}|proc=@var{p})

:: Set the flag "AutoReduce" to @var{s}.

@end table

Line 2746 Grobner bases. This is the default.

Line 1919 Grobner bases. This is the default.

@end itemize

/*&jp-texi

/*&ja

@node sm1_auto_reduce,,, SM1 Functions

@node sm1.auto_reduce,,, SM1 Functions

@subsection @code{sm1_auto_reduce}

@subsection @code{sm1.auto_reduce}

@findex sm1_auto_reduce

@findex sm1.auto_reduce

@table @t

@item sm1_auto_reduce(@var{s}|proc=@var{p})

@item sm1.auto_reduce(@var{s}|proc=@var{p})

:: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.

@end table

Line 2773 reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%

Line 1946 reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%

def sm1_slope(II,V,FF,VF) {

SM1_FIND_PROC(P);

A =[II,V,FF,VF];

sm1_push_int0(P,A);

sm1(P," slope toString ");

ox_check_errors2(P);

R = eval_str(sm1_pop(P));

return(R);

}

/*&en

/*&eg-texi

@node sm1.slope,,, SM1 Functions

@node sm1_slope,,, SM1 Functions

@subsection @code{sm1.slope}

@subsection @code{sm1_slope}

@findex sm1.slope

@findex sm1_slope

@table @t

@item sm1_slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})

@item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})

:: Returns the slopes of differential equations @var{ii}.

@end table

Line 2809 List (weight vector)

Line 1972 List (weight vector)

@end table

@itemize @bullet

@item @code{sm1_slope} returns the (geometric) slopes

@item @code{sm1.slope} returns the (geometric) slopes

of the system of differential equations @var{ii}

along the hyperplane specified by

the V filtration @var{v_filtration}.

Line 2829 of the slopes are returned.

Line 1992 of the slopes are returned.

/*&jp-texi

/*&ja

@node sm1_slope,,, SM1 Functions

@node sm1.slope,,, SM1 Functions

@subsection @code{sm1_slope}

@subsection @code{sm1.slope}

@findex sm1_slope

@findex sm1.slope

@table @t

@item sm1_slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})

@item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})

:: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.

@end table

Line 2854 of the slopes are returned.

Line 2017 of the slopes are returned.

@end table

@itemize @bullet

@item @code{sm1_slope} $B$O(B

@item @code{sm1.slope} $B$O(B

$BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}

$B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.

@item @var{v} $B$OJQ?t$N%j%9%H(B.

Line 2872 Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B,

Line 2035 Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B,

Slope $B$N@dBPCM$rLa$9(B.

/*&C-texi

/*&C

@example

[284] A= sm1_gkz([ [[1,2,3]], [-3] ]);

[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]);

[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]]);

[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]);

[287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);

@end example

/*&eg-texi

/*&en

@table @t

@item Reference

@code{sm_gb}

@code{sm.gb}

@end table

/*&jp-texi

/*&ja

@table @t

@item $B;2>H(B

@code{sm_gb}

@code{sm.gb}

@end table

/*&eg-texi

/*&en

@include sm1-auto-en.texi

@include sm1-auto.en

/*&jp-texi

/*&ja

@include sm1-auto-ja.texi

@include sm1-auto.ja

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