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 a1p1^3＋…＋a9p9^3=b is solvable with pj≤Pfor all 1≤j≤9. It is proved that one can take P=max（｜a1]…｜a9｜}^c ＋ ｜b｜^1/3 with c=2. This improves upon the earlier result with c=14 due to Liu （2013）.