On a form of degree $d$ in $2d+1$ variables ($d\geq 4$)

Abstract

For $k\geq 2$, we derive an asymptotic formula for the number of zeros of the forms $\prod_{i=1}^{k}(x_{2i-1}^2+x_{2i}^2)+\prod_{i=1}^{k}(x_{2k+2i-1}^2+x_{2k+2i}^2)-x_{4k+1}^{2k}$ and $x_1\prod_{i=1}^{k}(x_{2i}^2+x_{2i+1}^2)+x_{2k+2}\prod_{i=1}^{k}(x_{2k+2i+1}^2+x_{2k+2i+2}^2)-x_{4k+3}^{2k+1}$ in the box $1\leq x_i\leq P$ using the circle method.