Colloquium, Mathematical Sciences

Abstract: A height is a measure of the arithmetic complexity of a point. The essential minimum of a height function is the minimal value that the height function can attain at generic points. It is a difficult invariant to compute related to the equidistribution of small points and Bogomolov type conjectures. There are methods to compute upper and lower bounds for the essential minimum. In a joint work with Binggang Qu, Ricardo Menares, and Martin Sombra, we prove using linear programming techniques that the difference between the upper and lower bounds can be made arbitrarily small. Therefore, one can devise a theoretical algorithm to compute the essential minimum with arbitrary precision and thus the essential minimum is a "computable" real number. This result has applications in several classical problems like the integral Chebyshev constant of the unit interval, the spectrums of the Zhang-Zagier and the Faltings heights and the asymptotic behaviour of the length of the shortest vector in the lattice associated with the Grassmannian Gr(2,4).