Small prime solutions to cubic equations

Let a₁,..., a₉ be nonzero integers not of the same sign, and let b be an integer. Suppose that a₁,..., a₉ are pairwise coprime and a₁ + · · · + a₉ ≡ b (mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a₁,..., a₉, n) such that the cubic equation a₁p₁³ + · · · + a₉p₉³ = b is solvable with p_j ? P for all 1 ≤ j ≤ 9. It is proved that one can take P = max {|a₁|, ..., |a₉|}^c with c + |b|^1/3 with c = 2. This improves upon the earlier result with c = 14 due to Liu (2013).