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version 1.1, 2005/04/13 23:50:17

version 1.6, 2019/08/31 06:36:28

Line 1

/*$OpenXM$ */

/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.5 2012/11/28 05:07:31 takayama Exp $ */

/*&C

@c DO NOT EDIT THIS FILE

Line 18

$B7W;;Be?t4v2?$N$$$m$$$m$JITJQNL$N7W;;$,HyJ,:nMQAG$N7W;;$K5"Ce$9$k(B.

@code{sm1} $B$K$D$$$F$NJ8=q$O(B @code{OpenXM/doc/kan96xx} $B$K$"$k(B.

$B$J$*(B, sm1 server windows $BHG$O%P%$%J%jG[I[$7$F$$$J$$(B.

cygwin $B4D6-$G%=!<%9%3!<%I$+$i%3%s%Q%$%k$7(B, OpenXM/misc/packages/Windows

$B$K=>$$JQ99$r2C$($k$H(B sm1 $B%5!<%P$O(Bwindows $B$G$bF0:n$9$k(B.

$B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,

$BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.

$B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.

Line 49 to constructions in the ring of differential operators

Line 53 to constructions in the ring of differential operators

Documents on @code{sm1} are in

the directory @code{OpenXM/doc/kan96xx}.

The sm1 server for windows is not distributed in the binary form.

If you need to run it, compile it under the cygwin environment

following the Makefile in OpenXM/misc/packages/Windows.

All the coefficients of input polynomials should be

integers for most functions in this section.

Other functions accept rational numbers as inputs

Line 102 Grobner Deformations of Hypergeometric Differential Eq

Line 110 Grobner Deformations of Hypergeometric Differential Eq

* sm1.mul::

* sm1.distraction::

* sm1.gkz::

* sm1.mgkz::

* sm1.appell1::

* sm1.appell4::

* sm1.rank::

Line 502 x*dx+1

Line 511 x*dx+1

@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},needBack=@var{n})

:: 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})

Line 550 List

Line 559 List

Each polynomial is expressed as a string temporally for now.

When the optional variable @var{r} is set to one,

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

@item If you want to have a reduced basis or compute the initial form ideal exactly,

execute sm1.auto_reduce(1) before executing this function.

@item When the needBack option @var{n} is 1, it returns the answer is a different format as [groebner basis,[gb,1,all,[groebner basis, backward transformation]]]

@end itemize

/*&ja

Line 559 List

Line 571 List

@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},needBack=@var{n})

:: @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})

:: @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.

Line 600 List

Line 612 List

$B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.

$B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,

$BLa$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).

@item Reduced $B%0%l%V%J!<4pDl$^$?$O(B in_w $B$r7W;;$7$?$$$H$-$O(B, $B$3$N4X?t$N<B9T$NA0$K(B

sm1.auto_reduce(1) $B$r<B9T$7$F$*$/$3$H(B.

@item needBack $B%*%W%7%g%s$,(B 1 $B$N;~$O(B, $BB>$N>l9g$H$O0[$J$k7A<0(B

[groebner basis, [gb,1,all, [groebner basis, backward transformation]]]

$B$GEz$($rLa$9(B. (sm1 $B$N(B getAttribute $B$r;2>H(B)

@end itemize

/*&C

Line 697 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

Line 714 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]

@end example

/*&C

@example

[1834] sm1.gb([[dx^2-x,dx],[x]] | needBack=1);

[[[dx,dx^2-x,1],[dx,dx^2,1]],[gb,1,all,[[dx,dx^2-x,1],[[0,1],[1,0],[-dx,dx^2-x]]]]]

@end example

/*&en

@table @t

@item Reference

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

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

@end table

/*&ja

@table @t

@item $B;2>H(B

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

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

@end table

Line 1657 List

Line 1680 List

/*&en

@node sm1.mgkz,,, SM1 Functions

@subsection @code{sm1.mgkz}

@findex sm1.mgkz

@table @t

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

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

@var{A} and the weight @var{w} with the parameter vector @var{B}.

@end table

@table @var

@item return

List

@item p

Number

@item A, W, B

List

@end table

@itemize @bullet

@item Returns the modified GKZ hypergeometric system

(A-hypergeometric system) associated to the matrix

@item http://arxiv.org/abs/0707.0043

@end itemize

/*&ja

@node sm1.mgkz,,, SM1 Functions

@subsection @code{sm1.mgkz}

@findex sm1.mgkz

@table @t

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

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

@end table

@table @var

@item return

$B%j%9%H(B

@item p

$B?t(B

@item A, W, B

$B%j%9%H(B

@end table

@itemize @bullet

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

@item http://arxiv.org/abs/0707.0043

@end itemize

/*&C

@example

[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).

@end example

/*&en

@node sm1.appell1,,, SM1 Functions

@subsection @code{sm1.appell1}

Line 2002 not bihomogeneous.

Line 2089 not bihomogeneous.

Algorithm:

see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,

How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"

Note that the signs of the slopes are negative, but the absolute values

Note that the signs of the slopes s' are negative, but the absolute values -s'

of the slopes are returned.

In other words, when pF+qV is the gap, -s'=q/p is returned.

Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).

The number s satisfies 1<= s.

We have r=s-1=-1/s', and kappa=1/r=-s', which is used 1/Gamma(1+m*r) factor and exp(-tau^kappa)

in the Borel and Laplace transformations respectively.

Line 2046 Algorithm:

Line 2138 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"

$B$r$_$h(B.

Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,

Slope s' $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,

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

Slope $B$N@dBPCM(B -s' $B$rLa$9(B.

$B$D$^$j(B pF+qV $B$,(Bmicro$BFC@-B?MMBN$N(Bgap$B$G$"$k$H$-(B, -s'=q/p $B$rLa$9(B.

$B:G6a$NJ88%$G$O(B s=1-1/s' $B$r(B slope $B$H8F$s$G$$$k(B. $B2r$O(B O(s) $B$KB0$9$k(B.

$B?t(B s $B$O(B 1<= s $B$rK~$9(B.

r=s-1=-1/s' $B$*$h$S(B kappa=1/r=-s' $B$G$"$k(B.

$B$3$l$i$N?t$O(BBorel and Laplace $BJQ49$K$*$$$F$=$l$>$l(B 1/Gamma(1+m*r) factor,

exp(-tau^kappa) $B9`$H$7$F;H$o$l$k(B.

/*&C

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