Zeros of Systems of ${\mathfrak p}$-adic Quadratic Forms
  It is shown that a system of $r$ quadratic forms over a ${\mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$. 
D.R. Heath-Brown