Stephen Choi

Areas of Interest

My research interests are Diophantine approximations, Diophantine equations, Goldbach-Waring problems, polynomials with restricted coefficients and merit factors of binary sequences.

Solving Diophantine equations is one of the most fundamental and important problems in number theory. For example, the famous Fermat Last problem is to solve the Diophantine equation \( x^n + y^n = z^n\) for positive integer exponent \( n \) and integer unknowns \( x, y, z \). Another famous example is the Waring problem which investigates the integer solutions \( x_1, x_2, \ldots , x_s\) for the diagonal Diophantine equation \[ x_1^k + x_2^k + \cdots + x_s^k = b \] for constant integer \( b \ge 1\) and exponent \( k \ge 2\). In other words, the Waring problem studies representations of integers in a sum of fixed power of integers. Solving Diophantine equations over some special and interesting subsets of integers is also fascinating and attractive to number theorists. In particular, people are interested in considering prime solutions of Diophantine equations. The most famous problem of this type of study is the Goldbach problem which asks if every even integer greater than \( 5\) can be written as a sum of two primes, that is, to solve the Diophantine equations \( p_1 + p_2 = b\) for even integer \( b\) and prime unknowns \( p_1\) and \( p_2\). One area of my research work is considering the problem of the combination of both the Goldbach and Waring problems by studying the diagonal Diophantine equations \[ a_1p_1^k + a_2p_2^k + \cdots + a_sp_s^k = b \] where \( a_i\) are integer coefficients and \( p_i\) are prime unknowns. This is called the Goldbach-Waring problem. One of the goals in my research is to prove the solubility of the prime variables of the above Goldbach-Waring equations that do not grow too rapidly as the coefficients grow to infinity by using an analytic method such as the Hardy and Littlewood circle method.

Polynomials are fundamental objects in many branches in Mathematics. They pervade Mathematics. In particular, the study of polynomials with integer coefficients is fundamental and important and is always one of the core areas in number theory. Polynomials with coefficients restricted to some special and interesting subsets of integers are also fascinating and draw much interest. The simplest subset is the binary set \( \{ -1, 1\}\). In particular, people are interested in studying polynomials with coefficients \( +1\) or \( -1\). We call these polynomials Littlewood Polynomials. One can view Littlewood polynomials as finite binary sequences as the natural association of their coefficients to finite binary sequences. Undoubtedly, this kind of polynomials is also fascinating and attractive in other fields such as combinatorics and information theory, etc. The problems concern polynomials typically ask something about the size of the polynomials with an appropriate measure of the size and often with some restriction on the coefficients and degrees. We are particularly interested in studying the behaviour of the \( L_4\) norms over the unit circle of Littlewood polynomials. More specifically, one may want to know how small the \( L_4\) norm of Littlewood polynomials can be. This is still an open question if there is a non-trivial lower bound for the \( L_4\) norm of Littlewood polynomials of a given degree. This problem is closely related to the longstanding open problem raised by Littlewood and Erdös about the existence of the ultra-flat Littlewood polynomials. All these problems are still wide open. One of my research interests is to investigate and search for those Littlewood polynomials with very small \( L_4\) norms. This study is closely related to the problem of finding the so-called easily distinguishable binary sequences in information and communication theory.

Research Areas

• Number Theory