Macaulay2 » Documentation
Packages » BernsteinSato :: rationalFunctionSolutions
next | previous | forward | backward | up | index | toc

rationalFunctionSolutions -- rational solutions of a holonomic system



The rational solutions of a holonomic system form a finite-dimensional vector space. The only possibilities for the poles of a rational solution are the codimension one components of the singular locus. An algorithm to compute rational solutions is based on Gröbner deformations and works for ideals $I$ of PDE's - see the paper Polynomial and rational solutions of a holonomic system by Oaku, Takayama and Tsai (2000).

i1 : makeWA(QQ[x])

o1 = QQ[x, dx]

o1 : PolynomialRing, 1 differential variable(s)
i2 : I = ideal((x+1)*dx+5)

o2 = ideal(x*dx + dx + 5)

o2 : Ideal of QQ[x, dx]
i3 : rationalFunctionSolutions I

o3 = {-------------------------------}
       5     4      3      2
      x  + 5x  + 10x  + 10x  + 5x + 1

o3 : List


The most efficient method to find rational solutions of a system of differential equations is to find the singular locus, then try to find its irreducible factors. With these, call rationalFunctionSolutions(I, ff, w), where w should be generic enough so that the polynomialSolutions routine will not complain of a non-generic weight vector.

See also

Ways to use rationalFunctionSolutions :

For the programmer

The object rationalFunctionSolutions is a method function.