Simultaneous Cubic and Quadratic Diagonal Equations In 12 Prime Variables

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