GKZ hypergeometric system is a system of linear partial differential equations associated to (an integer -matrix of rank ) and . The book by Saito, Sturmfels and Takayama  discusses algorithmic methods to construct series solutions of the GKZ system. The current Asir-contrib-HG package is built in order to implement these algorithms. What we need for the implementation are mainly (1) Gröbner basis computation both in the ring of polynomials and in the ring of differential operators, and (2) enumeration of all the Gröbner bases of toric ideals. Asir and kan/sm1 provide functions for (1) and TiGERS provides a function for (2). These components communicate each other by OpenXM-RFC 100 protocol.
Let us see an example how to construct series solution of a GKZ hypergeometric system. The function dsolv_starting_term finds the leading terms of series solutions to a given direction.
 F = sm1_gkz( [ [[1,1,1,1,1], [1,1,0,-1,0], [0,1,1,-1,0]], [1,0,0]]); [[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1, -x4*dx4+x2*dx2+x1*dx1, -x4*dx4+x3*dx3+x2*dx2, -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4], [x1,x2,x3,x4,x5]]
 A= dsolv_starting_term(F,F, [1,1,1,1,0])$ 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]
 A; [[ 0 0 0 0 1 ]]  map(fctr,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]]]
Nobuki Takayama 2017-03-30