When does a system of quadratic forms $$Q_i(x)=q_{i1}x_1^2+\dotsb+q_{in}x_n^2,\ i\in[1,m]$$ have a nontrivial common zero? By the Chevalley–Warning theorem, $n>2m$ is a sufficient condition, but I need a more precise criterion; what happens if $n\le 2m$?
I am sure this has been studied; references or pointers will be appreciated.
The coefficients $q_{ij}$ of the forms $Q_i$ come from a field $K$, the finite fields of characteristic $3$ being of particular interest for me.
Unfortunately it seems to me that $n>2m$ is the best we can do in general. Let $Q(x,y)$ be a bivariate form that vanishes only trivially. For example, we can choose $Q$ to be the norm form of the quadratic extension of $K$.
It follows that for all the forms $$Q_i(x_1,\ldots,x_{2m}):=Q(x_{2i-1},x_{2i}),$$ $i=1,2,\ldots,m$, vanish simultaneously only when $x_1=x_2=\cdots=x_{2m}=0$.