ns_twistedlog.rr

ns_twistedlog.rr User’s Manual

Edition 1.0

February 2012

by Keisuke Nishitani

Copyright © Keisuke Nishitani 2012. All rights reserved.


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1 ns_twistedlog.rr


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1.1 ns_twistedlog.rr について

ns_twistedlog.rr’ は twisted logarithmic cohomology 群の計算, およびそれに基づいて, 多項式ベキの積分から定まるあるクラスの超幾何積分の満たす差分方程式系の計算と, 指数関数と多項式ベキの積分から定まるあるクラスの超幾何積分の満たす微分方程式系の計算を行うためのパッケージである.


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1.2 Twisted logarithmic cohomology 群の計算に関する関数


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1.2.1 ns_twistedlog.twisted_log_cohomology

ns_twistedlog.twisted_log_cohomology(FL,PL,VL)

:: Twisted logarithmic cohomology 群の middle cohomology 群の基底を返す.

FL

多項式のリスト

PL

パラメータのリスト

VL

変数のリスト

[1] ns_twistedlog.twisted_log_cohomology([x,y,1-x-y],[a,b,c],[x,y]);
-- nd_weyl_gr :0.003848sec(0.008291sec)
-- weyl_minipoly_by_elim :0.006988sec(0.007177sec)
-- generic_bfct_and_gr :0.01325sec(0.02175sec)
generic bfct : [[-1,1],[s,1],[s+a+b+c-1,1]]
S0 : 0
B_{S0 length : 1
-- fctr(BF) + base :0.001454sec(0.005543sec)
dimension : 1
[1]

[2] ns_twistedlog.twisted_log_cohomology([x,y,1-x-y],[-1,-2,-3],[x,y]);
-- nd_weyl_gr :0.001845sec(0.001838sec)
-- weyl_minipoly_by_elim :0.003972sec(0.003971sec)
-- generic_bfct_and_gr :0.007363sec(0.007584sec)
generic bfct : [[-1,1],[s,1],[s-7,1]]
S0 : 7
B_{S0 length : 36
-- fctr(BF) + base :0.02438sec(0.03323sec)
dimension : 3
[y^2*x^5,y^7,1]

[3] ns_twistedlog.twisted_log_cohomology([x*z+y,x^4+y^5+x*y^4],[0,0],[x,y,z]);
-- nd_weyl_gr :0.004sec(0.0028sec)
weyl_minipoly_by_elim : b-function does not exist
stopped in weyl_minipoly_by_elim2 at line 378 in file "/usr/local/ox/OpenXM/src/
asir-contrib/packages/src/nk_restriction.rr"
378                  error("weyl_minipoly_by_elim : b-function does not exist");
(debug)
参照

ns_twistedlog.twisted_log_cohomology(option)


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1.2.2 ns_twistedlog.twisted_log_cohomology(option)

ns_twistedlog.twisted_log_cohomology(...| exp = f, check = n, s0 = m, excp = v)

:: ns_twistedlog.twisted_log_cohomology のオプションの説明

f

多項式

n

0 または 1

m

整数

v

0 または 1

[4] ns_twistedlog.twisted_log_cohomology([x,y,1-x-y],[a,b,c],[x,y]|exp = x+y);
-- nd_weyl_gr :0.004sec + gc : 0.004sec(0.006156sec)
-- weyl_minipoly_by_elim :0sec(0.001558sec)
-- generic_bfct_and_gr :0.004sec + gc : 0.004sec(0.008213sec)
generic bfct : [[1,1],[s,1],[s-1,1],[s+a+b-1,1]]
S0 : 1
B_{S0 length : 3
-- fctr(BF) + base :0sec(0.000469sec)
dimension : 2
[y,1]

[5] ns_twistedlog.twisted_log_cohomology([x*z+y,x^4+y^5+x*y^4],[0,0],[x,y,z]|ch
eck = 1);
Hilbert polynomial : 1/24*x^4+65/12*x^3-529/24*x^2+727/12*x-51
holonomic : No
-- nd_weyl_gr :0.004001sec(0.002876sec)
weyl_minipoly_by_elim : b-function does not exist
stopped in weyl_minipoly_by_elim2 at line 378 in file "/usr/local/ox/OpenXM/src/
asir-contrib/packages/src/nk_restriction.rr"
378                 error("weyl_minipoly_by_elim : b-function does not exist");
(debug) 

[6] ns_twistedlog.twisted_log_cohomology([x*z+y,x^4+y^5+x*y^4],[0,0],[x,y,z]|s0 = 1
);
dimension : 3
[y,z,1]

[7] ns_twistedlog.twisted_log_cohomology([x,y,1-x-y],[a,b,c],[x,y]|excp = 1);
generic bfct : [[-1,1],[s,1],[s+a+b+c-1,1]]
S0 : 0
B_{S0 length : 1
dimension : 1
[[1],[a+b+c-1,a,b]]

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1.2.3 ns_twistedlog.difference_equation

ns_twistedlog.difference_equation(FL,PL,VL)

:: 多項式ベキの積分から定まる超幾何関数の満たす差分方程式系を返す.

FL

多項式のリスト

PL

多項式のベキのリスト(パラメータに対応)

VL

積分を行う変数のリスト

以下は, p(a,b,c) = \int_C x^{a-1}y^{b-1}(1-x-y)^{c-1} dxdyの満たす差分方程式系を計算した例である.

[8] ns_twistedlog.difference_equation([x,y,1-x-y],[a,b,c],[x,y]);              
-- nd_weyl_gr :0sec(0.000421sec)
-- weyl_minipoly_by_elim :0sec(0.001051sec)
Order : 1
[(-ea+1)*a-ea*b-ea*c,eb*a+(eb-1)*b+eb*c,ec*a+ec*b+(ec-1)*c]

以下のような入力に対しては正しく動かない.

[9] ns_twistedlog.difference_equation([x,y,1-x-y],[a,b,a-b],[x,y]);
-- nd_weyl_gr :0sec(0.0003741sec)
-- weyl_minipoly_by_elim :0.004sec + gc : 0.004sec(0.006554sec)
Order : 1
[-ea,eb,1]

[10] ns_twistedlog.difference_equation([x,y,1-x-y],[-a,-b,2*c],[x,y]);
-- nd_weyl_gr :0sec(0.0003951sec)
-- weyl_minipoly_by_elim :0sec(0.001059sec)
Order : 1
[(ea-1)*a+ea*b-2*ea*c,-eb*a+(-eb+1)*b+2*eb*c,ec*a+ec*b+(-2*ec+2)*c]
参照

ns_twistedlog.difference_equation(option)


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1.2.4 ns_twistedlog.difference_equation(option)

ns_twistedlog.difference_equation(... | exp = f, check = n, inhomo = h, shift = p,
order = m, excp = v)

:: ns_twistedlog.difference_equation のオプションの説明.

f

多項式

n

0 または 1

h

0 または 1

p

パラメータ

m

整数

v

0 または 1

[11] ns_twistedlog.difference_equation([x,y,1-x-y],[a,b,c],[x,y]|inhomo = 1);
-- nd_weyl_gr :0sec(0.0003991sec)
-- weyl_minipoly_by_elim :0sec(0.001058sec)
Order : 1
[[(-ea+1)*b*a-ea*b^2-ea*c*b,[((y^2-y)*dy+b*x+(b+c)*y-b)*dx+(-y^2+y)*dy^2+((-a-b-
c)*y+b)*dy,(-a-b-c)*x+(-b-c)*y]],[eb*a+(eb-1)*b+eb*c,[((y^2-y)*dy+b*x+(b+c)*y-b)
*dx+(-y^2+y)*dy^2+((-a-b-c)*y+b)*dy,-y]],[ec*b*a+ec*b^2+(ec-1)*c*b,[((y^2-y)*dy+
b*x+(b+c)*y-b)*dx+(-y^2+y)*dy^2+((-a-b-c)*y+b)*dy,(-a-b-c)*x-c*y]]]

[12] ns_twistedlog.difference_equation([x,y,1-x-y],[a,b,c],[x,y]|shift = a); 
-- nd_weyl_gr :0.004sec(0.0004289sec)
-- weyl_minipoly_by_elim :0sec(0.001042sec)
Order : 1
[(ea-1)*a+ea*b+ea*c]

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1.2.5 ns_twistedlog.differential_equation

ns_twistedlog.differential_equation(EXP,FL,PL,TVL,XVL)

:: 指数関数と多項式ベキの積分から定まる超幾何関数の満たす微分方程式系を返す.

EXP

多項式

FL

多項式のリスト

PL

多項式のベキのリスト

TVL

積分を行う変数のリスト

XVL

パラメータの変数のリスト

以下は f(x_1,x_2) = \int_C exp(x1*t1+x2*t2) t1^{a-1}*t2^{b-1}dt1dt2 の満たす微分方程式系を計算した例である.

[13] ns_twistedlog.differential_equation(x1*t1+x2*t2,[t1,t2],[a,b],[t1,t2],[x1,x2])
;    
-- nd_weyl_gr :0sec(0.0004089sec)
-- weyl_minipoly_by_elim :0sec(0.000495sec)
Order : 1
[x1*dx1+a,-x2*dx2-b]
参照

ns_twistedlog.differential_equation(option)


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1.2.6 ns_twistedlog.differential_equation(option)

ns_twistedlog.differential_equation(... | check = n, inhomo = h, diff = p,
order = m, excp = v)

:: ns_twistedlog.differential_equation のオプションの説明

n

0 または 1

h

0 または 1

p

パラメータ

m

整数

v

0 または 1

[14] ns_twistedlog.differential_equation(x1*t1+x2*t2,[t1,t2],[a,b],[t1,t2],[x1,x2]|
diff = x1);
-- nd_weyl_gr :0sec(0.0007901sec)
-- weyl_minipoly_by_elim :0sec + gc : 0.008sec(0.006175sec)
Order : 1
[x1*dx1+a]

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1.3 その他の関数


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1.3.1 ns_twistedlog.twisted_deRham

ns_twistedlog.twisted_deRham(F,P,VL)

:: Twisted de Rham cohomology 群の middle cohomology 群の基底を返す.

F

多項式

P

パラメータ

VL

変数のリスト

[15] ns_twistedlog.twisted_deRham(x*y*(1-x-y),a,[x,y]);                                        
-- nd_weyl_gr :0sec(9.489e-05sec)
-- weyl_minipoly :0sec(0.0002191sec)
-- generic_bfct_and_gr :0sec(0.000423sec)
generic bfct : [[1,1],[s,1]]
S0 : 0
B_{S0 length : 1
-- fctr(BF) + base :0sec(6.008e-05sec)
dimension : 0
[]

[16] ns_twistedlog.twisted_deRham(x*y*(1-x-y),-1,[x,y]);
-- nd_weyl_gr :0sec(0.0001891sec)
-- weyl_minipoly :0sec(0.000247sec)
-- generic_bfct_and_gr :0sec(0.0006139sec)
generic bfct : [[1,1],[s,1],[s-1,1]]
S0 : 1
B_{S0 length : 3
-- fctr(BF) + base :0.004sec(0.0002241sec)
dimension : 3
[x,y,1]

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1.3.2 ns_twistedlog.holonomic

ns_twistedlog.holonomic(Id, VL, DVL)

:: D の左イデアル Id がホロノミックならば標準単項式のリストを返す. ホロノミックでないならば-1を返す.

Id

イデアルの生成元のリスト

VL

変数のリスト

DVL

変数のリスト (VL に対応する微分作用素の方の変数)

[17] ns_twistedlog.holonomic([x*dy,y*dx],[x,y],[dx,dy]);                          
Hilbert polynomial : x^2+1
holonomic : Yes
holonomic rank : 1
[1]

[18] ns_twistedlog.holonomic([(x^3-y^2)*dx+3*x^2,(x^3-y^2)*dy-2*y],[x,y],[dx,
dy]);
Hilbert polynomial : 1/2*x^3+2*x^2+1/2*x+2
holonomic : No
-1

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Index

??:   N  
?????  ?

N
ns_twistedlog.difference_equation 1.2.3 ns_twistedlog.difference_equation
ns_twistedlog.difference_equation(option) 1.2.4 ns_twistedlog.difference_equation(option)
ns_twistedlog.differential_equation 1.2.5 ns_twistedlog.differential_equation
ns_twistedlog.differential_equation(option) 1.2.6 ns_twistedlog.differential_equation(option)
ns_twistedlog.twisted_deRham 1.3.1 ns_twistedlog.twisted_deRham
ns_twistedlog.twisted_log_cohomology 1.2.1 ns_twistedlog.twisted_log_cohomology
ns_twistedlog.twisted_log_cohomology(option) 1.2.2 ns_twistedlog.twisted_log_cohomology(option)

??:   N  

[??] [??] [???] [ ? ]

??


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