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Undergraduate publishes paper

## Undergraduate student publishes paper

• Date10 October 2023

BSc student Murray Child and Professor Widmer publish paper in Research Journal

The paper "On Mahler's inequality and small integral generators of totally complex number fields" accepted for publication by Acta Arithmetica is based on a third year project by Murray Child. The project was suggested and supervised by Prof Martin Widmer.

The Mahler measure is a function from complex polynomials (or rather, its leading coefficient, and its roots, considered as points on the complex plane) to the reals: it has valuable applications across various fields, such as Number Theory, Group Theory and Mathematical Physics.

The eponymous mathematician Kurt Mahler had already provided a sharp lower bound for the Mahler measure (in terms of the degree and discriminant of a polynomial) back in 1964. Professor Widmer suggested to investigate whether the  bound can be improved for a restricted class of polynomials: monic polynomials where all roots with modulus greater than 1 come in complex conjugate pairs. (The final version of the paper is slightly more generalised than this, but the proof derives from the same principles.) Professor Widmer had already shown this for m=3 pairs of complex conjugate roots, and the project aim was to extend this to larger values, and perhaps even to all positive integers m.

Murray Child says: "Since I was attempting to prove a novel result, I really didn’t know whether the project would work at all! But 22 hand-written pages later, after much toiling over the summer, I had my first full proof for generalised m – and just like an inefficient sorting algorithm, this went through many iterations. The final version is maybe the 8th or 9th major rewrite.

I must extend my huge gratitude to my co-author Martin Widmer for his extraordinary help with this paper. He was responsible not only for bringing the question to my attention and providing the groundwork, but also for the rewriting of my dissertation to an appropriate format for publication, and for helping me understand and write about the important applications of the main result, like providing lower bounds for the Mahler measure of minimal polynomials of integral generators, and complementary upper bounds, derived via the Minkowski-embedding. He has been a very organised, precise and knowledgeable person to work with on this project."

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