On the Hasse principle for a block-diagonal form and a form of lower degree

We investigate the Hasse principle for systems consisting of one form of degree $k$ which decomposes as a sum of $n$-ary forms, and one general form of degree $d < k$. By exploiting the block structure of the first form before applying Weyl differencing, we refine the method of Brandes and Parsell (arXiv:2003.04350) in this setting; in particular, for a sum of binary quartic forms and a quadratic form, the requirement $s > 516$ is relaxed to $s > 96$, which also improves on the bound $s > 100$ available for general forms.