<p>The paper considers an optimization problem of constructing a polynomial structure for solutions of polyharmonic equations. A triharmonic equation in the Cartesian coordinates with three non-trivial boundary conditions is taken as a basis. The solution of the considered boundary-value problem is proved to belong to the class of positive operators. It is shown that polynomial expansions for the boundary values of the triharmonic function near the boundary of the upper half-plane generate the Taylor formula of the triharmonic Poisson integral. Based on the obtained result, the existence of additional boundary conditions for solutions of triharmonic equations in the Cartesian coordinates is proved.</p>