Simultaneous cubic and quadratic diagonal equations in 12 prime variables

The system of equations $$\begin{aligned}&u_1p_1^2 + \cdots u_sp_s^2 = 0,\\&v_1p_1^3 v_sp_s^3 0 \end{aligned}$$ has prime solutions $$(p_1, \ldots , p_s)$$ for $$s \ge 12$$ assuming that the modulo each p. This is proved via Hardy–Littlewood circle method, building on Wooley’s work corresponding over integers and recent results Vinogradov’s mean value theorem. Additionally, a set sufficient conditions local solvability given: If both are solvable 2, quadratic equation 3, p at least 7 $$u_i$$ $$v_i$$ not zero p, then