Diophantine Equations and the Theory of Computation: A lecture series

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• Prof Minhyong Kim, International Centre for Mathematical Sciences, Edinburgh
• Wednesday 21 May 2025, 14:00-15:30
• The Computer Laboratory, FW26 and Online.

If you have a question about this talk, please contact Challenger Mishra.

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A Diophantine equation, named after the Egyptian mathematician Diophantus of Alexandria (c. 2C), is a polynomial equation

f(x_1, x_2, ... , x_n) = 0

with integer coefficients whose solutions with rational or integer coordinates are the objects of interest. The computation of such solution sets comprises some of the oldest problems in mathematics, and any given case has a strong tendency to be surprisingly difficult even when approached with all the machinery that modern arithmetic geometry has at its disposal.

This series of lectures will focus on the computation of the full rational solution set for two-variable equations:

f(x, y) = 0.

To get a sense of why this might be difficult, consider the case of Fermat’s equation:

(x/z)^{n +(y/z)}n =1

where it took over 350 years to be sure that the obvious solutions were the only ones. (Some equations in three variables can be effectively reduced to the two-variable case.) Writing down a random equation like

y^{3 = x}6 23x⁵ 37x4 691x^{3 – 631204x}2 5169373941

and noting that it has the solution (1, 1729), it is very hard to know if it has any other solutions.

As of now, the study of Diophantine equations has a curious tendency to touch on an incredibly broad range of mathematics, including algebraic/differential geometry, algebraic topology, global analysis, and mathematical physics.

Equations in two variables include elliptic curves (degree 3), out of whose study abstract and far-reaching frameworks for number theory and algebraic geometry have emerged, such as a conjectural category of Grothendieck/Voevodsky motives, with mysterious connections to automorphic forms and the Langlands programme.

This represents a remarkable confluence of theory and computation: about 60 years ago, as modern scientific computation was first making its way into mathematics, Birch and Swinnerton-Dyer started examining elliptic curves from an algorithmic viewpoint in the Mathematical Laboratory at Cambridge and formulated their famous conjecture. The BSD conjecture is a powerful tool for computational arithmetic geometry, in that it can be used to compute solution sets even if the conjecture itself is not known to be true.

These lectures will review some of this history and the current status of the theory with an algorithmic focus. Along the way, if time allows, some general reflections on the interaction of number theory and computation will be presented, including undecidability, complexity, quantum computation, and the universality of the theory of Diophantine equations.

One main goal will be to encourage interest in hyperbolic equations, i.e., two-variable equations of degree larger than 3, including the so-called ‘non-abelian Chabauty theory’ of algorithmic resolution. With the advent of machine learning, I believe the time is right for this extension of the BSD philosophy to be brought to the attention of scientists with diverse backgrounds in the theory of computation. (In particular, do not worry if some of the terms in this abstract are unfamiliar to you.)

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