I was wondering whether there exists any result of the form

"if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta n$ for some explicit $\delta$, then $f(x) = 0$ satisfies the Hasse principle."

I'm particularly interested in the case $n = 4$. Thanks!