Summary: Let \(k\) be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers \(h^{i,j}(X)=\dim H^j(X,\Omega_X^i)\) that hold for every smooth proper variety \(X\) over \(k\) are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of \(X\) are given by the span of the \(h^{i,0}(X)\) and the \(h^{0,j}(X)\) (and their duals \(h^{i,n}(X)\) and \(h^{n,j}(X))\). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.