Equal values of standard counting polynomials

The following discrete geometrical question provides a background for some classical diophantine problems. For given positive integers m, n, can an m-dimensional and an n-dimensional unit cube, simplex, pyramid or octahedron contain equally many integral points? Apart from some trivial cases, the question leads to 9 families of diophantine equations, see Table 1. In this paper we give a brief survey of known results on these equations, and prove some new theorems concerning the solutions.