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

Abstract

For $k\ge 2$, 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 $1\le x_i\le P$ using the circle method.

arXiv comment12 pages