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 `$(OpenXM_HOME)/lib/asir/contrib-packages'.
The system sm1
is a system to compute in the ring of differential
operators.
Many constructions of invariants
in the computational algebraic geometry reduce
to constructions in the ring of differential operators.
Documents on sm1
are in
the directory OpenXM/doc/kan96xx
.
All the coefficients of input polynomials should be integers for most functions in this section. Other functions accept rational numbers as inputs and it will be explicitely noted in each explanation of these functions.
[283] sm1.deRham([x*(x-1),[x]]); [1,2]
The author of sm1
: Nobuki Takayama, takayama@math.sci.kobe-u.ac.jp
The author of sm1 packages : Toshinori Oaku, oaku@twcu.ac.jp
Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
Grobner Deformations of Hypergeometric Differential Equations,
1999, Springer.
http://www.math.kobe-u.ac.jp/KAN
ox_sm1_forAsir
Serverox_sm1_forAsir
sm1
server for asir
.
ox_sm1_forAsir
is the sm1
server started from asir
by the command sm1.start
.
In the standard setting, ox_sm1_forAsir
=
`$(OpenXM_HOME)/lib/sm1/bin/ox_sm1'
+
`$(OpenXM_HOME)/lib/sm1/callsm1.sm1' (macro file) $(LOAD_SM1_PATH)
,
$(OpenXM_HOME)/lib/sm1
,
/usr/local/lib/sm1
in this order.
sm1
macros.
sm1.start
ox_sm1_forAsir
on the localhost.
ox_sm1_forAsir
on the localhost.
It returns the descriptor of ox_sm1_forAsir
.
Xm_noX = 1
to start ox_sm1_forAsir
without a debug window.
ord
. For example,
when you are working in the
ring of differential operators on the variable x
and dx
(dx
stands for
),
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
.
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
).
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
ox_launch
, sm1.push_int0
, sm1.push_poly0
,
ord
sm1.sm1
sm1
server to execute the command string s.
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]
sm1.start
, ox_push_int0
, sm1.push_poly0
, sm1.get_Sm1_proc()
.
sm1.push_int0
type(f)
is 2 (recursive polynomial),
f is converted to a string (type == 7)
and is sent to the server by ox_push_cmo
.
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.
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]
ox_push_cmo
sm1.gb
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].
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,
[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
[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>>]]]
sm1.reduction
, sm1.rat_to_p
sm1.deRham
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.
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.
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]
sm1.start
, deRham
(sm1 command)
sm1.hilbert
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
sm1.start
, sm1.gb
, longname
sm1.genericAnn
rest
(v).
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]
sm1.start
sm1.wTensor0
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.
[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]]
sm1.reduction
sm1.reduction_noH
is for the Weyl algebra.
[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]]
sm1.start
, d_true_nf
sm1.xml_tree_to_prefix_string
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))
om_*
, OpenXM/src/OpenMath
, eval_str
sm1.syz
[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]]]]
sm1.mul
sm1.mul_h
is for homogenized Weyl algebra.
sm1.mul(p0*dp0,1,[p0])
returns
dp0*p0+1
.
A variable order such that d-variables 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]]
sm1.distraction
sm1
server to compute the distraction of f.
sm1
server of the descriptor number p
to compute the distraction of f in the ring of differential
operators with variables v.
[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
distraction2(sm1)
,
sm1.gkz
[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]]
sm1.mgkz
[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).
sm1.appell1
[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
sm1.appell4
[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
sm1.rank
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
sm1.auto_reduce
sm1.slope
sm1.slope
returns the (geometric) slopes
of the system of differential equations ii
along the hyperplane specified by
the V filtration v_filtration.
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]);
sm.gb
sm1.ahg
sm1.bfunction
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);
sm1.call_sm1
sm1.ecart_homogenize01Ideal
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);
sm1.ecartd_gb
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);
sm1.ecartd_gb_oxRingStructure
sm1.ecartd_isSameIdeal_h
Example:
input II=[(1-x)^2*dx+h*(1-x)]$ JJ = [(1-x)*dx+h]$ V=[x]$ sm1.ecartd_isSameIdeal_h([II,JJ,V]);
sm1.ecartd_reduction
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);
sm1.ecartd_reduction_noh
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);
sm1.ecartd_syz
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);
sm1.gb_oxRingStructure
sm1.gb_reduction
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);
sm1.gb_reduction_noh
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);
sm1.generalized_bfunction
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]);
sm1.isSameIdeal_in_Dalg
Example:
Input1 II=[(1-x)^2*dx+(1-x)]$ JJ = [(1-x)*dx+1]$ V=[x]$ sm1.isSameIdeal_in_Dalg(II,JJ,V);
sm1.laplace
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);
sm1.restriction
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]);
sm1.saturation
Example:
sm1.saturation([[x2^2,x2*x4, x2, x4^2], [x2,x4], [x2,x4]]);
sm1.ahg
sm1.bfunction
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);
sm1.call_sm1
sm1.ecart_homogenize01Ideal
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);
sm1.ecartd_gb
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);
sm1.ecartd_gb_oxRingStructure
sm1.ecartd_isSameIdeal_h
Example:
input II=[(1-x)^2*dx+h*(1-x)]$ JJ = [(1-x)*dx+h]$ V=[x]$ sm1.ecartd_isSameIdeal_h([II,JJ,V]);
sm1.ecartd_reduction
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);
sm1.ecartd_reduction_noh
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);
sm1.ecartd_syz
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);
sm1.gb_oxRingStructure
sm1.gb_reduction
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);
sm1.gb_reduction_noh
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);
sm1.generalized_bfunction
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]);
sm1.isSameIdeal_in_Dalg
Example:
Input1 II=[(1-x)^2*dx+(1-x)]$ JJ = [(1-x)*dx+1]$ V=[x]$ sm1.isSameIdeal_in_Dalg(II,JJ,V);
sm1.laplace
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);
sm1.restriction
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]);
sm1.saturation
Example:
sm1.saturation([[x2^2,x2*x4, x2, x4^2], [x2,x4], [x2,x4]]);
@vfill @eject
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