Sm1 OX Server マニュアル

Edition : auto generated by oxgentexi on October 24, 2017

OpenXM.org

@overfullrule=0pt

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1 SM1 函数

この節では sm1 の ox サーバ ox_sm1_forAsir とのインタフェース関数を解説する. これらの関数はファイル ‘sm1.rr’ で定義されている. ‘sm1.rr’ は ‘$(OpenXM_HOME)/lib/asir-contrib’ にある. システム sm1 は微分作用素環で計算するためのシステムである. 計算代数幾何のいろいろな不変量の計算が微分作用素の計算に帰着する. sm1 についての文書は OpenXM/doc/kan96xx にある.

なお, sm1 server windows 版はバイナリ配布していない. cygwin 環境でソースコードからコンパイルし, OpenXM/misc/packages/Windows に従い変更を加えると sm1 サーバはwindows でも動作する.

とこに断りがないかぎりこの節のすべての関数は, 有理数係数の式を入力としてうけつけない. すべての多項式の係数は整数でないといけない.

[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


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1.1 ox_sm1_forAsir サーバ


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1.1.1 ox_sm1_forAsir

ox_sm1_forAsir

:: asir のための sm1 サーバ.


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1.2 函数一覧


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1.2.1 sm1.start

sm1.start()

:: localhost で ox_sm1_forAsir をスタートする.

return

整数

[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
参照

ox_launch, sm1.push_int0, sm1.push_poly0, ord


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1.2.2 sm1.sm1

sm1.sm1(p,s)

:: サーバ sm1 にコマンド列 s を実行してくれるようにたのむ.

return

なし

p

s

文字列

[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]
参照

sm1.start, ox_push_int0, sm1.push_poly0, sm1.get_Sm1_proc().


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1.2.3 sm1.push_int0

sm1.push_int0(p,f)

:: オブジェクト f を識別子 p のサーバへ送る.

return

なし

p

f

オブジェクト

[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


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1.2.4 sm1.gb

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

:: v 上の微分作用素環において f のグレブナ基底を計算する.

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

:: v 上の微分作用素環において f のグレブナ基底を計算する. 結果を分散多項式のリストで戻す.

return

リスト

p, q, r

f, v, w

リスト

[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]]

上の例において,

[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]]

上の例において二つのモノミアル

[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>>]]]
参照

sm1.auto_reduce, sm1.reduction, sm1.rat_to_p


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1.2.5 sm1.deRham

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

:: 空間 C^n - (the zero set of f=0) のドラームコホモロジ群の次元を計算してくれるようにサーバに頼む.

return

リスト

p

f

文字列 または 多項式

v

リスト

[332] sm1.deRham([x^3-y^2,[x,y]]);
[1,1,0]
[333] sm1.deRham([x*(x-1),[x]]);
[1,2]
参照

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.


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1.2.6 sm1.hilbert

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

:: 多項式の集合 f のヒルベルト多項式を計算する.

hilbert_polynomial(f,v)

:: 多項式の集合 f のヒルベルト多項式を計算する.

return

多項式

p

f, v

リスト

[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
参照

sm1.start, sm1.gb, longname


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1.2.7 sm1.genericAnn

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

:: f^s のみたす微分方程式全体をもとめる. v は変数のリストである. ここで, s は v[0] であり, f は変数 rest(v) 上の多項式である.

return

リスト

p

f

多項式

v

リスト

[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]
参照

sm1.start


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1.2.8 sm1.wTensor0

sm1.wTensor0([f,g,v,w]|proc=p)

:: fg の D-module としての 0 次テンソル積を 計算する.

return

リスト

p

f, g, v, w

リスト

[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]]

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1.2.9 sm1.reduction

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

::

return

リスト

f

多項式

g, v, w

リスト

p

数 (ox_sm1 のプロセス番号)

[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]]
参照

sm1.start, d_true_nf


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1.2.10 sm1.xml_tree_to_prefix_string

sm1.xml_tree_to_prefix_string(s|proc=p)

:: XML で書かれた OpenMath の木表現 s を前置記法になおす.

return

String

p

Number

s

String

[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))
参照

om_*, OpenXM/src/OpenMath, eval_str


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1.2.11 sm1.syz

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

:: v 上の微分作用素環において f の syzygy を計算する.

return

リスト

p

f, v, w

リスト

[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]]]]

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1.2.12 sm1.mul

sm1.mul(f,g,v|proc=p)

:: sm1サーバ に f かける gv 上の微分作用素環でやってくれるように頼む.

return

多項式またはリスト

p

f, g

多項式またはリスト

v

リスト

[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]]

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1.2.13 sm1.distraction

sm1.distraction([f,v,x,d,s]|proc=p)

:: sm1f の distraction を計算してもらう.

return

リスト

p

f

多項式

v,x,d,s

リスト

[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
参照

distraction2(sm1),


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1.2.14 sm1.gkz

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

:: 行列 A とパラメータ B に付随した GKZ 系 (A-hypergeometric system) をもどす.

return

リスト

p

A, B

リスト

[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]]


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1.2.15 sm1.mgkz

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

:: 行列 A, weight W とパラメータ B に付随した modified GKZ 系 (A-hypergeometric system) をもどす.

return

リスト

p

A, W, B

リスト

[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).

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1.2.16 sm1.appell1

sm1.appell1(a|proc=p)

:: F_1 または F_D に対応する方程式系を戻す.

return

リスト

p

a

リスト

[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



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1.2.17 sm1.appell4

sm1.appell4(a|proc=p)

:: F_4 または F_C に対応する方程式系を戻す.

return

リスト

p

a

リスト

[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


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1.2.18 sm1.rank

sm1.rank(a|proc=p)

:: 微分方程式系 a の holonomic rank を戻す.

return

p

a

リスト

[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


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1.2.19 sm1.auto_reduce

sm1.auto_reduce(s|proc=p)

:: フラグ "AutoReduce" を s に設定.

戻り値

p

s


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1.2.20 sm1.slope

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

:: 微分方程式系 ii の slope を戻す.

return

p

ii

リスト (方程式)

v

リスト (変数)

f_filtration

リスト (weight vector)

v_filtration

リスト (weight vector)

Algorithm: "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" をみよ. Slope s’ の本来の定義では, 符号が負となるが, このプログラムは, Slope の絶対値 -s’ を戻す. つまり pF+qV がmicro特性多様体のgapであるとき, -s’=q/p を戻す. 最近の文献では s=1-1/s’ を slope と呼んでいる. 解は O(s) に属する. 数 s は 1<= s を満す. r=s-1=-1/s’ および kappa=1/r=-s’ である. これらの数はBorel and Laplace 変換においてそれぞれ 1/Gamma(1+m*r) factor, exp(-tau^kappa) 項として使われる.

[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]);


参照

sm.gb


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1.2.21 sm1.ahg

sm1.ahg(A)

: It idential with sm1.gkz(A).


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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);

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1.2.23 sm1.call_sm1

sm1.call_sm1(F)

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


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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);


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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.

Note. Functions in the category ecart changes the global environment in the sm1 server. If you interrupted these functions, run sm1.ecartd_gb with a small input and terminate it normally. Then, the global environment is reset to the normal state. Reference. G. Granger, T. Oaku, N. Takayama, Tangent cone algorithm for homogeized differential operators, 2005.

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);
 input3
   F=[[x^2,y+x],[x+y,y^3], [2*x^2+x*y,y+x+x*y^3]]$
   FF=[F,"x,y",[[dx,1,dy,1],[x, -1, y, -1,dx, 1, dy, 1]],
             ["degreeShift",[[0,1],[-3,1]]]]$
   sm1.ecartd_gb(FF);

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1.2.26 sm1.ecartd_gb_oxRingStructure

sm1.ecartd_gb_oxRingStructure()

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


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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]);

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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. When the output is G, G[3] is F and G[0]-(G[1]*A-sum(k,G[2][k]*G[3][k]))=0 holds. F must be (0,1)-hohomogenized (see sm1.ecart_homogenize01Ideal). This function does not check if the given order is admissible for the ecart reduction. To do this check, use sm1.ecartd_gb.

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]]]$
   G=sm1.ecartd_reduction(dx+dy,FF);
   G[0]-(G[1]*(dx+dy)+G[2][0]*F[0]+G[2][1]*F[1]);

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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);

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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);

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1.2.31 sm1.gb_oxRingStructure

sm1.gb_oxRingStructure()

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


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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);

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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);

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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]);

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1.2.35 sm1.integration

sm1.integration(I,V,R)

: It computes the integration 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 integrations from 0 to d are computed. Note that, in case of vector input, INTEGRATION 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.integration(I,V,R | degree=key0)

: This function allows optional variables degree

Algorithm:

T.Oaku and N.Takayama, math.AG/9805006, http://www.arxiv.org

Example:

 sm1.integration([dt - (3*t^2-x), dx + t],[t,x],[t]);
   The output [[n0,F0],[n1,F1],...] means that H^0=D^n0/F0, H^(-1)=D^n1/F1, ...
   The free basis of the vector space D^n is denoted by e0, e1, ... 

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1.2.36 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);

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1.2.37 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);

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1.2.38 sm1.rat_to_p

sm1.rat_to_p(F)

: It returns the denominator of F and the numerator of F. They are returned in a list. All elements of the denominator and numerator are polynomial objects with integer coefficients. Note that dn and nm do not regard rational numbers as a factional object and this function is necessary to send data to sm1, which accept only integers and does not accept rational numbers.

Example:

     sm1.rat_to_p(1/2*x+1);
       [x+2,2]
     sm1.rat_to_p([1/2*x,1/3*x]);
       [[3*x,2*x],6]

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1.2.39 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.lanl.gov

Example:

 sm1.restriction([dx^2-x,dy^2-1],[x,y],[y]);
   The output [[n0,F0],[n1,F1],...] means that H^0=D^n0/F0, H^(-1)=D^n1/F1, ...
   The free basis of the vector space D^n is denoted by e0, e1, ... 

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1.2.40 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]]);

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1.2.41 sm1.ahg

sm1.ahg(A)

: It idential with sm1.gkz(A).


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1.2.42 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);

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1.2.43 sm1.call_sm1

sm1.call_sm1(F)

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


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1.2.44 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);


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1.2.45 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.

Note. Functions in the category ecart changes the global environment in the sm1 server. If you interrupted these functions, run sm1.ecartd_gb with a small input and terminate it normally. Then, the global environment is reset to the normal state. Reference. G. Granger, T. Oaku, N. Takayama, Tangent cone algorithm for homogeized differential operators, 2005.

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);
 input3
   F=[[x^2,y+x],[x+y,y^3], [2*x^2+x*y,y+x+x*y^3]]$
   FF=[F,"x,y",[[dx,1,dy,1],[x, -1, y, -1,dx, 1, dy, 1]],
             ["degreeShift",[[0,1],[-3,1]]]]$
   sm1.ecartd_gb(FF);

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1.2.46 sm1.ecartd_gb_oxRingStructure

sm1.ecartd_gb_oxRingStructure()

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


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1.2.47 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]);

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1.2.48 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. When the output is G, G[3] is F and G[0]-(G[1]*A-sum(k,G[2][k]*G[3][k]))=0 holds. F must be (0,1)-hohomogenized (see sm1.ecart_homogenize01Ideal). This function does not check if the given order is admissible for the ecart reduction. To do this check, use sm1.ecartd_gb.

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]]]$
   G=sm1.ecartd_reduction(dx+dy,FF);
   G[0]-(G[1]*(dx+dy)+G[2][0]*F[0]+G[2][1]*F[1]);

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1.2.49 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);

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1.2.50 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);

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1.2.51 sm1.gb_oxRingStructure

sm1.gb_oxRingStructure()

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


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1.2.52 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);

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1.2.53 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);

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1.2.54 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]);

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1.2.55 sm1.integration

sm1.integration(I,V,R)

: It computes the integration 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 integrations from 0 to d are computed. Note that, in case of vector input, INTEGRATION 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.integration(I,V,R | degree=key0)

: This function allows optional variables degree

Algorithm:

T.Oaku and N.Takayama, math.AG/9805006, http://www.arxiv.org

Example:

 sm1.integration([dt - (3*t^2-x), dx + t],[t,x],[t]);
   The output [[n0,F0],[n1,F1],...] means that H^0=D^n0/F0, H^(-1)=D^n1/F1, ...
   The free basis of the vector space D^n is denoted by e0, e1, ... 

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1.2.56 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);

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1.2.57 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);

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1.2.58 sm1.rat_to_p

sm1.rat_to_p(F)

: It returns the denominator of F and the numerator of F. They are returned in a list. All elements of the denominator and numerator are polynomial objects with integer coefficients. Note that dn and nm do not regard rational numbers as a factional object and this function is necessary to send data to sm1, which accept only integers and does not accept rational numbers.

Example:

     sm1.rat_to_p(1/2*x+1);
       [x+2,2]
     sm1.rat_to_p([1/2*x,1/3*x]);
       [[3*x,2*x],6]

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1.2.59 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.lanl.gov

Example:

 sm1.restriction([dx^2-x,dy^2-1],[x,y],[y]);
   The output [[n0,F0],[n1,F1],...] means that H^0=D^n0/F0, H^(-1)=D^n1/F1, ...
   The free basis of the vector space D^n is denoted by e0, e1, ... 

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1.2.60 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]]);

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Index

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Index Entry  Section

H
hilbert_polynomial 1.2.6 sm1.hilbert

O
ox_sm1_forAsir 1.1.1 ox_sm1_forAsir

S
sm1.ahg 1.2.21 sm1.ahg
sm1.ahg 1.2.41 sm1.ahg
sm1.appell1 1.2.16 sm1.appell1
sm1.appell4 1.2.17 sm1.appell4
sm1.auto_reduce 1.2.19 sm1.auto_reduce
sm1.bfunction 1.2.22 sm1.bfunction
sm1.bfunction 1.2.42 sm1.bfunction
sm1.call_sm1 1.2.23 sm1.call_sm1
sm1.call_sm1 1.2.43 sm1.call_sm1
sm1.deRham 1.2.5 sm1.deRham
sm1.distraction 1.2.13 sm1.distraction
sm1.ecartd_gb 1.2.25 sm1.ecartd_gb
sm1.ecartd_gb 1.2.45 sm1.ecartd_gb
sm1.ecartd_gb_oxRingStructure 1.2.26 sm1.ecartd_gb_oxRingStructure
sm1.ecartd_gb_oxRingStructure 1.2.46 sm1.ecartd_gb_oxRingStructure
sm1.ecartd_isSameIdeal_h 1.2.27 sm1.ecartd_isSameIdeal_h
sm1.ecartd_isSameIdeal_h 1.2.47 sm1.ecartd_isSameIdeal_h
sm1.ecartd_reduction 1.2.28 sm1.ecartd_reduction
sm1.ecartd_reduction 1.2.48 sm1.ecartd_reduction
sm1.ecartd_reduction_noh 1.2.29 sm1.ecartd_reduction_noh
sm1.ecartd_reduction_noh 1.2.49 sm1.ecartd_reduction_noh
sm1.ecartd_syz 1.2.30 sm1.ecartd_syz
sm1.ecartd_syz 1.2.50 sm1.ecartd_syz
sm1.ecart_homogenize01Ideal 1.2.24 sm1.ecart_homogenize01Ideal
sm1.ecart_homogenize01Ideal 1.2.44 sm1.ecart_homogenize01Ideal
sm1.gb 1.2.4 sm1.gb
sm1.gb_d 1.2.4 sm1.gb
sm1.gb_oxRingStructure 1.2.31 sm1.gb_oxRingStructure
sm1.gb_oxRingStructure 1.2.51 sm1.gb_oxRingStructure
sm1.gb_reduction 1.2.32 sm1.gb_reduction
sm1.gb_reduction 1.2.52 sm1.gb_reduction
sm1.gb_reduction_noh 1.2.33 sm1.gb_reduction_noh
sm1.gb_reduction_noh 1.2.53 sm1.gb_reduction_noh
sm1.generalized_bfunction 1.2.34 sm1.generalized_bfunction
sm1.generalized_bfunction 1.2.54 sm1.generalized_bfunction
sm1.genericAnn 1.2.7 sm1.genericAnn
sm1.gkz 1.2.14 sm1.gkz
sm1.hilbert 1.2.6 sm1.hilbert
sm1.integration 1.2.35 sm1.integration
sm1.integration 1.2.55 sm1.integration
sm1.isSameIdeal_in_Dalg 1.2.36 sm1.isSameIdeal_in_Dalg
sm1.isSameIdeal_in_Dalg 1.2.56 sm1.isSameIdeal_in_Dalg
sm1.laplace 1.2.37 sm1.laplace
sm1.laplace 1.2.57 sm1.laplace
sm1.mgkz 1.2.15 sm1.mgkz
sm1.mul 1.2.12 sm1.mul
sm1.push_int0 1.2.3 sm1.push_int0
sm1.rank 1.2.18 sm1.rank
sm1.rat_to_p 1.2.38 sm1.rat_to_p
sm1.rat_to_p 1.2.58 sm1.rat_to_p
sm1.reduction 1.2.9 sm1.reduction
sm1.restriction 1.2.39 sm1.restriction
sm1.restriction 1.2.59 sm1.restriction
sm1.saturation 1.2.40 sm1.saturation
sm1.saturation 1.2.60 sm1.saturation
sm1.slope 1.2.20 sm1.slope
sm1.sm1 1.2.2 sm1.sm1
sm1.start 1.2.1 sm1.start
sm1.syz 1.2.11 sm1.syz
sm1.syz_d 1.2.11 sm1.syz
sm1.wTensor0 1.2.8 sm1.wTensor0
sm1.xml_tree_to_prefix_string 1.2.10 sm1.xml_tree_to_prefix_string

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