This project concerns Galois theory and algebraic number theory. Understanding all the words in the following description is not a pre-requisite for participating.
Let f(x) be an irreducible, monic polynomial with integer coefficients. The Mahler measure of the polynomial is a real number M(f)>=1, defined as the absolute value of the product of those roots of f(x) which lie outside the unit circle in the complex plane. A classic theorem of Kronecker states (under the assumptions above) that M(f) = 1 if and only if f(x) is a cyclotomic polynomial. A famous still-open conjecture of D. H. Lehmer from 1933 is that there exists a universal constant c>1 such that if f is not cyclotomic, then M(f) >= c. It is widely conjectured that c can in fact be taken to be the single root outside the unit circle of the famous polynomial appearing in Lehmer’s original paper:
P(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1; this root is approximately 1.176.
C.J. Smyth in 1971 proved the Lehmer conjecture for all polynomials which are not reciprocal. The polynomial f(x) is reciprocal if its coefficients read the same forward as backward (as in the polynomial P above), which is equivalent to saying that alpha is a root of f if and only if 1/alpha is. Therefore, only reciprocal polynomials are of interest in studying the Lehmer conjecture. This condition of being reciprocal restricts the Galois group to be strictly smaller than the full symmetric group. The degree of the f(x) must be an even integer 2n, and the Galois group will be a subgroup of a certain permutation group of order 2^n * n!, which is a wreath product of a group of order 2 and S_n. On the other hand, an important result of Amoroso and David from 2006 shows the Lehmer conjecture holds if the Galois group is very small (say bounded by any polynomial in n). So we have existing upper and lower bounds on the Galois groups of polynomials which might be counterexamples to Lehmer’s conjecture. We would like to explore the possibility of improving either of these bounds.
The first goals of this project are:
- To learn important background material.
- Write code (probably using sage or magma) to compute the Mahler measure and Galois groups of many polynomials.
- Formulate some refined conjectures based on the data.
If we make a lot of progress on these goals, it may be possible to spend some time and effort trying to prove new results along these lines.
Project Mentor: Bobby Grizzard
Graduate Student Mentor: John Yin
Team Members: Hernan Aldabe, Erkin Delic, Matthew Norton, Sammy Ross