This paper deals with the problem of interpolating partial functions over finite distributive lattices by lattice polynomial functions. More precisely, this problem can be formulated as follows: Given a finite dis-tributive lattice L and a partial function f from D ⊆ L n to L, find all the lattice polynomial functions that interpolate f on D. If the set of lattice polynomials interpolating a function f is not empty, then it has a unique upper bound and a unique lower bound. This paper presents a new description of these bounds and proposes an algorithm for computing them that runs in polynomial time, thus improving existing methods. Furthermore, we present an empirical study on randomly generated datasets that illustrates our theoretical results.