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

الأصل · EN

For k≥ 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 and x₁∏ᵢ₌₁ᵏ(x₂ᵢ²+x₂ᵢ₊₁²)+x₂ₖ₊₂∏ᵢ₌₁ᵏ(x₂ₖ₊₂ᵢ₊₁²+x₂ₖ₊₂ᵢ₊₂²)-x₄ₖ₊₃²ᵏ⁺¹ in the box 1≤ xᵢ≤ P using the circle method.