Assume that $V$ is a finite dimensional real vector space of dimension $n$.
Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple $(x\_1,x\_2,\ldots,x\_k)$ with $\sum\_{i=1}^k x\_i =0$?



Non alternative $k$-linear maps vanishing on $\sum x_i=0$
**No.** Notice that by linearity we should have
$$
 f(v\_1,\dots,v\_{k-1},\alpha\_1v\_1+\dots+\alpha\_{k-1}v\_{k-1})
 =\frac{f(\alpha\_1v\_1,\dots,\alpha\_{k-1}v\_{k-1},-\alpha\_1v\_1-\dots-\alpha\_{k-1}v\_{k-1})}{(-1)^k\alpha\_1\cdots\alpha\_{k-1}}
 =0
$$
for all nonzero $\alpha\_1,\dots,\alpha\_{k-1}$. Since each linear combination is a sum of two linear combinations with nonzero coefficients, this also holds for all $\alpha\_i$ [this argument works also for all fields with at least 3 elements]. Thus, the linear map $f$ vanishes on all linearly dependent $k$-tuples, in which case it is well known to be alternative.


