Cyclotomic points on curves. (English) Zbl 1029.11009

Bennett, M. A. (ed.) et al., Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21-26, 2000. Natick, MA: A K Peters. 67-85 (2002).
Let \(f(x,y)\) be a Laurent polynomial with complex coefficients, and let \(V(f)\) be the area of its Newton polytope. A pair \((a,b)\) is said to be a cyclotomic point if both \(a\) and \(b\) are roots of unity. Suppose that \(f\) is such that there are only finitely many cyclotomic points \((a,b)\) for which \(f(a,b)=0.\) The authors prove that then on the curve \(f(x,y)=0\) there are at most \(22 V(f)\) cyclotomic points. They give an example of \[ f(x,y)=xy+1/xy+x+1/x+y+1/y+1 \] which shows that the constant \(22\) cannot be replaced by a constant smaller than \(16.\) An infinite family of polynomials for which the constant is approximately \(10\) is also constructed. Finally, the authors give a sharp version of their upper bound for the number of cyclotomic points on a curve. Since one needs extra work to compute this sharp bound, the bound \(22 V(f)\) seems more practical for applications. In the paper, the authors also give an algorithm for finding the cyclotomic part of a polynomial in one variable and the literature where one can find much more general (but not so sharp) results.
For the entire collection see [Zbl 1002.00005].


11C08 Polynomials in number theory
11R09 Polynomials (irreducibility, etc.)