On a form of degree d in 2d+12d+1 variables (d≥4d≥4)

For k≥2k≥2, we derive an asymptotic formula for the number of zeros of the forms∏i=1k(x2i−12+x2i2)+∏i=1k(x2k+2i−12+x2k+2i2)−x4k+12k andx1∏i=1k(x2i2+x2i+12)+x2k+2∏i=1k(x2k+2i+12+x2k+2i+22)−x4k+32k+1 in the box 1≤xi≤P1≤xi≤P using the circle method.