Binomial binary forms are homogeneous polynomials with integer coefficients in two variables of the form $ax^d+by^d$. For $d=2$, these are quadratic forms, the study of which goes back to Brahmagupta and Fermat (the so called Pell's equation). Sums of two squares have been investigated by Ramanujan and Landau. An asymptotic estimate for the number of representation of an integer by a positive definite quadratic form has been obtained by Bernays. For $d\ge 3$, the fundamental result of Thue on diophantine approximation yields, for $m\not=0$ and assuming that $a/b$ is not a $d$-th power, the finiteness of the set of pairs of integers $(x,y)$ such that $ax^d+by^d=m$. Among the many mathematicians who contributed to the study of representation of integers by binomial binary forms, we may quote Mike Bennett, Roger Heath Brown, Neil Dummigan, Christopher Hooley, Christopher Skinner and Trevor Wooley, until Cam Stewart and Stanley Yao Xiao found a very general theorem on the representation of integers by binary forms, producing a best possible asymptotic estimate. They gave a completely explicit version for binomial binary forms. With Étienne Fouvry we recently investigated the number of integers which are represented by one form in a family of binary forms. In this lecture I will present the special case of binomial binary forms.

Bio: Michel Waldschmidt is a French mathematician. He earned his doctorate from the University of Bordeaux in 1972 under the supervision of Jean Fresnel. He taught in the University Paris 6 (Pierre et Marie Curie, now Sorbonne University) from 1972 to 2012, when he became Emeritus Professor. His research has largely concerned number theory, in particular transcendental numbers and Diophantine approximation.