A distribution property of the solutions of a congruene modulo a large prime (Q2544234)

Let \(Z\) denote the domain of ordinary integers, and let \(f(X_1,\ldots,X_n)\in Z[X_1,\ldots,X_n]\) be a homogeneous polynomial of degree \(d\ge 2\) in the \(n\ge 2\) indeterminates \(X_1,\ldots,X_n\). It is proved that if \(f\) is absolutely irreducible modulo a suitably large prime \(p\) and \[f(x_1,\ldots,x_n) \equiv 0\pmod p\tag{*}\] has at least \(\frac12 p^{n-1}\) solutions \(x = (x_1,\ldots,x_n)\), then every subcube \[ S(i_1,\ldots, i_n) = \{x\mid i_j\mu \le x_j< (i_j+1)\mu,\ j=1,\ldots,n\},\ i_1,\ldots,i_n = 0,1,2\ldots,\lambda - 1,\] where \(\lambda= [p^{1/n}/10d]\), \(\mu=p/\lambda\), contains a solution of (*).