Shuntaro Yamagishi

Solving polynomial equations in many variables in primes

Solving polynomial equations in primes is a fundamental problem in number theory. For example, the twin prime conjecture can be phrased as the statement that the equation x_1 - x_2 - 2 = 0 has infinitely many solutions in primes. Let F \in Z[x_1, ..., x_n] be a degree d homogeneous form. In 2014, Cook and Magyar proved the existence of prime solutions to the equation F(x_1, ..., x_n) = 0 under certain assumptions on F. In particular, their result requires the number of variables n to be an exponential tower in d. I will talk about a result related to this work of Cook and Magyar improving on the number of variables required.