  In this paper, the intersection of bivariate orthogonal polynomials on
triangle patches will be investigated. The result is interesting by its own but
also has important applications in the theory of a posteriori error estimation
for finite element discretizations with $p$-refinement, i.e., if the local
polynomial degree of the test and trial functions is increased to improve the
accuracy. A triangle patch is a set of disjoint open triangles whose closed
union covers a neighborhood of the common triangle vertex. On each triangle we
consider the space of orthogonal polynomials of degree n with respect to the
weight function which is the product of the barycentric coordinates. We show
that the intersection of these polynomial spaces is the null space. The
analysis requires the derivation of subtle representations of orthogonal
polynomials on triangles. Up to four triangles have to be considered to
identify that the intersection is trivial.
