Research Overview

My research is in number theory, principally the classical theory of Diophantine equations and systems — the constructive study of integer and rational solutions of polynomial equations. Much of my work follows the elementary, constructive tradition of Fermat and Euler: given a family of equations, or a combinatorial or geometric constraint on integers, I seek explicit parametric families of solutions or prove that none can exist. My work over more than three decades clusters around the following interconnected themes:

• Equal sums of like powers and the Tarry–Escott problem. A large part of my work addresses the classical Prouhet–Tarry–Escott problem and its generalizations — finding disjoint sets of integers whose sums of k-th powers agree for several consecutive values of k. I have obtained new ideal solutions of the Tarry–Escott problem of degrees four, five, seven and eleven, developed new methods for constructing multigrade chains, and — sometimes, in joint work with Jarosław Wróblewski — produced symmetric Diophantine systems and triads or families of integers with equal sums of squares, cubes and higher powers.

• Symmetric and parametric equations of the type f(x,y) = f(u,v). Another recurring theme is the systematic solution of quartic, quintic and sextic equations of the shape f(x,y) = f(u,v), and related symmetric systems. I have developed quadratic-form and matrix-theoretic methods for reducing such equations to more tractable equations, and applied these both to construct new families of solutions and, in several cases, to settle questions of solvability completely. 

• Configurations of integers and rational points with prescribed power-sum or geometric properties. Many of my papers construct integers or rational numbers satisfying constraints expressed through squares, cubes or biquadrates — sets whose sumset consists entirely of perfect squares, sextuples whose pairwise sums are squares, or biquadrates whose sum is a perfect square — and extend naturally into Diophantine geometry: rational quadrilaterals and cyclic polygons with integer sides and areas, points at rational distances on conics and parabolas, and rational points in arithmetic or geometric progression on circles and Huff curves.

• Elliptic curves, surfaces, and families of high rank. Several papers, including joint work with Andrew Bremner, Maciej Ulas and Arman Shamsi Zargar, construct families of elliptic curves of high rank arising from symmetric Diophantine systems, and diagonal quartic and sextic surfaces carrying infinitely many rational points; related work examines the Fermat cubic and quartic curves over cyclic fields, and octic Diophantine equations tied to families of elliptic curves.

• Matrix analogues of classical Diophantine problems. A distinctive strand of my research transposes classical Diophantine questions — the Tarry–Escott problem, the Fermat equation, root extraction — into the setting of matrices, including the extraction of n-th roots of 2×2 matrices, the algebraic solution of unilateral matrix polynomial equations, and matrix analogues of multigrade chains.

• Algebraic structure of solution sets. In work partly with Kenneth Nwabueze and Omar Kihel, I have studied when the integer solutions of a linear or parametrized quadratic Diophantine equation can be endowed with a natural group structure, connecting elementary Diophantine analysis with abstract algebra.

• Related classical questions. My early work with Andrzej Schinzel examined the number of terms in the irreducible factors of a polynomial over Q, and I have also written on Fermat's Last Theorem and some exponential Diophantine equations.

Across this range of topics, my approach is largely elementary and constructive — favouring explicit parametrizations, algebraic identities and careful case analysis over deep machinery — in keeping with the long Indian tradition of Diophantine analysis. I continue to work and publish results related to these problems.