As a natural (and expectable) extension of [my earlier question](https://mathoverflow.net/questions/54619/when-does-pa-b0-for-a-ne-b-ensure-p00):



> 
> How large must be a set $A\subset F\_2^n$ to ensure that if $P$ is a *cubic* polynomial in $n$ variables over the field $F\_2$, vanishing at every non-zero point of the sumset $2A:=\{a\_1+a\_2\colon a\_1,a\_2\in A\}$, then also $P(0)=0$? 
> 
> 
> 


(Here $n$ is a given positive integer, to be thought of as a growing parameter.)




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Ultimately, I want to know how large must $A$ be for every given degree $\deg P$. Say, if $P$ has degree zero, then, trivially, $|A|\ge 2$ suffices. Furthermore, it is easy to see that for $P$ linear, one needs $|A|\ge 3$ (while $|A|\ge 2$ is insufficient). For $P$ quadratic, it [suffices](https://mathoverflow.net/questions/54619/when-does-pa-b0-for-a-ne-b-ensure-p00) to have $|A|\ge n+3$. In the case where $P$ is cubic, at least $|A|>2n$ is needed: consider, for instance, the set 
 $$ A=\{0,e\_1,...,e\_n,e\_1+e\_2,...,e\_1+e\_n\}, $$ 
where $e\_i$ are the vectors of the standard basis, and the polynomial 
 $$ P=\sum\_{1<i<j\le n} x\_1x\_ix\_j+\sum\_{1\le i<j\le n} x\_ix\_j+\sum\_{1\le i\le n} x\_i+1. $$
Must $|A|$ actually be quadratic (or, perhaps, even exponential) in $n$? (If $|A|>2^{7n/8}$, then $A$ contains an affine $4$-dimensional subspace; hence $2A$ contains a linear $4$-dimensional subspace, and the rest follows with a minor effort.) 



When does $P(a−b)=0$  for $a≠b$  ensure $P(0)=0$? (Continued.)
About three years after posting this question, the answer is now known (although there still can be some room for improvements). Namely, Lemma 1 from a [recent joint paper](https://arxiv.org/abs/1605.01506) by Ernie Croot, Peter Pach, and myself reads as follows:



> 
> Suppose that $n\ge 1$ and $d\ge 0$ are integers, $P$ is a multilinear
>  polynomial in $n$ variables of total degree at most $d$ over a field
>  $\mathbb F$, and $A\subseteq{\mathbb F}^n$ is a set with $|A|>2\sum\_{0\le i\le d/2}\binom ni$.
>  If $P(a-b)=0$ for all $a,b\in A$ with $a\ne b$, then also $P(0)=0$.
> 
> 
> 


In particular, for $P$ cubic it suffices to have $|A|\ge 2n+3$ to conclude that $P(0)=0$. This is essentially sharp: an example above shows that at least $|A|\ge 2n+1$ is needed. 


