The Hilbert–Kunz function conjecture for the nodal cubic

Let $f=x^{3}+y^{3}+xyz$ be the element of $(\mathbb{Z}/2)[x,y,z]$ defining a nodal cubic. Let $\phi_f$ denote the associated function, and let $\phi_0$ be the function defined recursively by the magnification rules in the source. The Hilbert–Kunz function conjecture.

$$\phi_f=t+\phi_0.$$

This conjectural formula gives the values of $\phi_f$ at rational points and, as explained in the source, implies a precise value for the Hilbert–Kunz multiplicity of $uv+f$.