# integrate rational function over a polyhedral domain

Hi,

I would like to integrate (symbolically or numerically) a rational function on some polyhedral domain. As an example, my function would be:

sage: f(a,b,c) = 1/((a+b)^2 * c)


and my domain would be the unbounded polyhedron:

sage: I = [
....: [0,1,0,0],     # a > 0
....: [0,0,1,0],     # b > 0
....: [0,0,0,1],     # c > 0
....: [-1,1,1,1],    # a + b + c > 1
....: [1,-1,-1,0],   # a + b < 1
....: [0,0,-1,1]     # b < c
....: ]
sage: P = Polyhedron(ieqs=I)


Is it possible to do it in Sage? With some optional package? With some other software?

(Note for suspicious minds: the integral is finite.)

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## Comments

Sage has something for polynomial functions (via the latte_int optional package), see

but I'm not aware of something for more general rational functions.

( 2020-09-14 03:53:03 +0200 )edit

The distinction between polynomial and rational function matters a lot: the integration of a polynomial over a polytope is a rational number (that can be computed exactly by latte_int). With rational function, the result is very often transcendental and could also be infinite.

( 2020-09-16 13:43:30 +0200 )edit