We say that a field $K$ is $C\_{m}$ if it satisfies the following property: for every positive integer $n$ and every sequence of positive integers $(d\_{1},\dotsc,d\_{r})$ satisfying $d\_{1}^{m} + \dotsb + d\_{r}^{m} \le n$, every sequence of $(F\_{1},\dotsc,F\_{r})$ of homogeneous polynomials in $K[x\_{0},\dotsc,x\_{n}]$ with $\deg F\_{i} = d\_{i}$ has a common nontrivial zero in $K^{n+1} \setminus \{0\}$.



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> Are local fields $C\_{2}$?
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Motivation: I was reading the paper "Period-index bounds for arithmetic threefolds" ([link](https://arxiv.org/abs/1704.05489)) by Antieau, Auel, Ingalls, Krashen, Lieblich; (one case of) their version of the period-index conjecture (Conjecture 1.3) says that $\mathrm{ind} | \mathrm{per}^{n+1}$ for classes in $\operatorname{Br}(K)$ where $K$ is a field of transcendence degree $n$ over a local field $k$. For $n=0$, this is by Albert, Brauer, Hasse, Noether; for $n=1$, this is by Saltman and Parimala-Suresh; the authors of the above paper address the $n=2$ case for $p$-adic local fields and prove the slightly weaker bound $\mathrm{ind} | \mathrm{per}^{4}$ for classes whose period is prime to $6p$. Another version of the period-index conjecture (see [here](https://arxiv.org/abs/math/0511244)) says that if $K$ is a $C\_{m}$-field then $\mathrm{ind} | \mathrm{per}^{m-1}$ for every class in $\operatorname{Br}(K)$, and I was wondering if the first conjecture can be viewed as a special case of the second.



Are local fields $C_{2}$?
No: see Guy Terjanian, "Un contre-example à une conjecture d'Artin", *C. R. Acad. Sci. Paris Sér. A–B* **262** (1966) A612 for an example of homogeneous form of degree $4$ in $18$ variables over the $2$-adics that has non non-trivial zero, showing that $\mathbb{Q}\_2$ is not $C\_2$ (Artin had conjectured that the $\mathbb{Q}\_p$ were $C\_2$).


**PS:** On the other hand, the function fields of curves over finite fields *are* $C\_2$: I'm not sure whether this is due to Lang or Nagata (and there may be subtleties between $C\_2$ and $C'\_2$), but see Serre, *Galois Cohomology* (2d ed, 1995), chapter II, §4.5, remark (b).


