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Singular vectors of a representation of a finite-dimensional simple Liealgebra are weight vectors in the underlying module that are nullified bypositive root vectors. In this article, we use partial differential equationsto find all the singular vectors of the polynomial representation of the simpleLie algebra of type $F_4$ over its basic irreducible module. As applications,we obtain a new combinatorial identity about the number 24 and explicitgenerators of invariants. Moreover, we show that the number of irreduciblesubmodules contained in the space of homogeneous harmonic polynomials withdegree $k\geq 2$ is $\geq [|k/3|]+[|(k-2)/3|]+2$.

Polynomial Representation of $F_4$ and a New Combinatorial Identity about Twenty-Four