Polyharmonic equations, to my understanding, are defined as:
$$\Delta ^k u = 0$$
i.e. one repeatedly applies the laplace operator to the function a certain number of times and the result must be 0.
The system can be discretized quite easily using finite differences or in the case I care about, triangle meshes, using the discrete Laplace-Beltrami operator.
However, this system is incomplete, we must also have boundary conditions, else there is a trivial solution yielded by $u = 0$.
So, let's say I am given a mesh $M$. I can construct a matrix $L$ through the use of the laplace beltrami oerator. But this is only possible at interior (not boundary) vertices.
Let's say I know which vertices in my mesh are fixed, what do I do to construct an appropriate linear system to solve the polyharmonic problem in the discrete setting?