Suppose that $f\in \mathbb{Z}[x\_1,\dots,x\_n]$ and $f$ is a homogenous polynomial of degree $d$. Can we always construct $f$ such that the hypersurface $S\_f=\{x \in \mathbb{Z}^n:f(x)=0\}$ exhibits the failure of the Hasse principle? In particular, I am trying to understand why the non-singularity condition in results of Birch (Forms in Many Variables), Browning-Prendiville etc. is vital. I am told that there in case of $d=4$, there is an example of Hasse failure, probably by Swinnerton-Dyer ($S\_f$ for $f(x\_1,x\_2,x\_3,x\_4)=7x\_1^4+8x\_2^4-9x\_3^4-14x\_4^4$) (not sure how it is a Hasse failure and how easy it is to prove that it is one, but at any rate, it is a non-singular hypersurface that demonstrates that 4 variables are not nearly enough). Can we find more examples of such hypersurfaces which fail to satisfy the hassy principle, for degree $d=4$ or otherwise? I was just hoping to find a large class of examples of such $f$, not just one.



Examples of non-singular hypersurfaces exhibiting Hasse principle failures
Daniel Loughran's suggestion to study the paper [Random Diophantine Equations by Poonen and Voloch](https://math.mit.edu/%7Epoonen/papers/random.pdf) was a good one.


Te conjecture 3.2 of the paper Daniel Loughran linked, together with Appendix A (both due to Colliot-Thélène), implies that the Hasse principle holds for nonsingular hypersurfaces of degree $4$ in $n$ variables as soon as $n>4$.


So, conjecturally, the condition $n>48$ in Browning-Prendiville is not at all necessary, and the $n>48$ condition is "merely" a restriction of the technique.


For $n=4$, there are more counterexamples. The paper [Counterexamples to the Hasse principle in families](https://doi.org/10.1017/S0004972721000885) contains two infinite families along the lines of Swinnerton-Dyer's example.


