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# A program to find the staring terms of the asymptotic expansions of a given regular holonomic system

1.
Starting asir with the OpenXM extensions. (You can get this system from http://www.math.kobe-u.ac.jp/OpenXM. )
bash-2.03$asir This is Risa/Asir, Version 20000126. Copyright (C) FUJITSU LABORATORIES LIMITED. 1994-1999. All rights reserved. xm version 20000202. Copyright (C) OpenXM Developing Team. 2000. ox_help(0); ox_help("keyword"); ox_grep("keyword"); for help message Loading ~/.asirrc  load("dsolv"); 1  2. Computing the GKZ hypergeometric system associated with the matrix and .  F = sm1_gkz( [ [[1,1,1,1,1],[1,1,0,-1,0],[0,1,1,-1,0]], [-1,0,0]])$

3.
Finding the staring terms to the direction

w = (1,1,1,1,0)

by using the Algorithm 2.3.14 of .

A = dsolv_starting_term(F,F,[1,1,1,1,0])$ The output is as follows:  Computing the initial ideal. Done. Computing a primary ideal decomposition. Primary ideal decomposition of the initial Frobenius ideal to the direction [1,1,1,1,0] is [[[x5+2*x4+x3+1, x5+3*x4-x2+1, x5+2*x4+x1+1, 3*x5^2+(8*x4+6)*x5+8*x4+3, x5^2+2*x5-8*x4^2+1, x5^3+3*x5^2+3*x5+1], [x5+1,x4,x3,x2,x1]]] ----------- root is [ 0 0 0 0 -1 ] ----------- dual system is [x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2 +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2, x4-2*x3+3*x2-2*x1, x5-x3+x2-x1, 1]  4. The four staring terms are as follows:  A = map(myfctr,A); [[[1/8,1],[x5,-1],[log(x2)+log(x4)-2*log(x5),1], [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5),1]], [[1,1],[x5,-1],[-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]], [[1,1],[x5,-1],[-log(x1)+log(x2)-log(x3)+log(x5),1]], [[1,1],[x5,-1]]]  5. Draw a graph of four starting terms on the line x1 = x, x2 = x, x3 = x, x4 = x, x5 = 1( ). See Figure 1.  B=map(subst,A,x1,x,x2,x,x3,x,x4,x,x5,1); /* Do not use t */ gnuplot_plot_function(B); Nobuki Takayama $BJ?@.(B12$BG/(B2$B7n(B7\$BF|(B