Theoretical Statistics and Mathematics Unit, ISI Delhi

September 13, 2017 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Michel Waldschmidt,
University of Jussieu, Paris, France

Title:
Representation of positive integers by binary cyclotomic forms

Abstract of Talk

The homogeneous form $\Phi_n(X,Y)$ of degree $\varphi(n)$ which is associated with the cyclotomic polynomial $\phi_n(X)$ is dubbed {\it cyclotomic binary form}.
A positive integer $k\ge 1$ is said to be {\it representable by a cyclotomic binary form} if there exist integers $n,x,y$ with $n\ge 3$ and $\max\{|x|, |y|\}\ge 2$ such that $\Phi_n(x,y)=k$. We prove that the set of integers which are representable by a cyclotomic binary form has density zero. For $k$ an integer which is representable by a cyclotomic binary form, we prove that the number of such representations is finite; let $a_k$ be the number of $(n,x,y)$ with $n\ge 3$, $\max\{|x|, |y|\}\ge 2$ and $\Phi_n(x,y)=k$. The average of the nonzero values of the sequence $(a_k)_{k\ge 1}$ grows like $\sqrt{\log k}$.