Diophantine equations in semiprimes

Abstract

A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, \ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to that of the seminal work of B. J. Birch, for which the equation $F (x_1, \ldots, x_n) = 0$ has infinitely many solutions whose coordinates are all semiprimes. Previously it was known due to \'A. Magyar and T. Titichetrakun that under the same hypotheses there exist infinite number of integer solutions to the equation whose coordinates have at most $384 n^{3/2} d (d+1)$ prime factors. Our main result reduces this bound on the number of prime factors from $384 n^{3/2} d (d+1)$ to $2$.