Ten unknowns for Hilbert's tenth problem over the integers

Hilbert’s Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation $P(z_1,\dots ,z_{10})=0$ (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.