\begin{equation*}
N(\boldsymbol{F};P)=c_{\boldsymbol{F}}\cdot P^{n-d_1-d_2}+O(P^{n-d_1-d_2-\delta}),\ \text{for some }\delta>0,
\end{equation*}
where the constant $c_{\boldsymbol{F}}$ is the product of local densities. We note that this asymptotic formula agrees with the Manin-Peyre conjecture.

Compared to the previous work, we improve the lower bound $n_0(\boldsymbol{F})$ in most cases, with the exception of case $d_2-d_1=1$. In particular, if $F_1$ and $F_2$ are non-singular forms, then we obtain  $$n_0(\boldsymbol{F})=3(d_2-1)2^{d_2-1}+(d_1-1)2^{d_1},$$ provided that $d_2\geq 5d_1$ and $d_1$ is large enough. This yields a substantial improvement over the previous bound $n_0(\boldsymbol{F})=(d_1+2)(d_2-1)2^{d_2-1}+(d_1-1)2^{d_1-1}$.

To achieve this, we develop a new differencing argument together with the van der Corput differencing argument, delivering an efficient upper-bound estimate for mean values of exponential sums associated with two forms in many variables of different degrees, when the difference between degrees is sufficiently large. Furthermore, the method described in this paper is flexible enough to apply to forms in many variables of differing degrees in general. This is joint work with Trevor Wooley.