### Philadelphia Area Number Theory Seminar

Wednesday, December 13, 2017 - 2:15pm

#### Robert Grizzard

Lafayette College

We’ll recall what the Weil height is, and how it can be viewed as a norm on the multiplicative group. We’ll then explain some approximation results in this norm from a recent joint work with J.D. Vaaler. The simplest such result (which has a “2-line” proof*) states that if an algebraic point is “close” to points deﬁned over some ﬁeld, a power of it is already deﬁned over that ﬁeld; this can be made eﬀective. Then we will review the Lehmer conjecture (“roots of irreducible integer polyno-mials can’t all be too close to the unit circle unless they are all on it”) and various generalizations recently put forth by Ga¨el R´emond, and use our multiplicative ap-proximation results to say some things about these conjectures. *If you write kind of small.