Integral and Rational Solutions of Polynomial Equations

Given a polynomial equation with rational coefficients, understanding when the equation has an integral/rational solution is a fundamental problem in arithmetic geometry and number theory. In this proposal, I will explore two important aspects of this problem: “Polynomial equations in primes” and “Polynomials represented by norm forms in function fields”. Solving diophantine equations in primes is a fundamental problem in number theory. For example, one of the most famous conjectures in number theory is the twin prime conjecture which states that there exist infinitely many solutions (x_1,x_2) with prime coordinates to the equation x_1–x_2=2. There has been a significant breakthrough recently on solving general linear equations in primes due to Green, Tao and Ziegler. However, for general higher degree equations progress has been limited. In the first part of this proposal, I will develop a novel approach to this problem based on the circle method and a generalization of the hyperbola method to improve our understanding of prime solutions to general polynomial equations. Understanding the local to global principle for general algebraic varieties is an important aspect in arithmetic geometry and number theory, because it deepens our understanding of the arithmetic nature of the equations involved. One area which has attracted great interest is understanding the Hasse principle and weak approximation of the equation P(t)=N(x_1,…,x_n), where N(x_1,…,x_n) is a norm form and P(t) is a polynomial with rational coefficients. While there has been a significant progress in this area, the progress over function fields has been limited. Understanding the function field analogues of problems can lead to further developments of the corresponding problems over number fields. In the second part of this proposal, I will follow the historical development of this problem over number fields and establish the function field analogues of these results.