Building on work of Dubickas and Smyth regarding the metric Mahler measure and the authors regarding extremal norms associated to the Mahler measure, the authors introduce a new set of norms associated to the Mahler measure of algebraic numbers which allow for an equivalent reformulation of problems like the Lehmer problem and the Schinzel-Zassenhaus conjecture on a single spectrum. We present ...

For / R 1 odd, this result has been proved by Shult [31]. The purpose of this paper is to prove Theorem 1. I when j Sz I is even. We remark that the groups listed in Theorem 1 .l all satisfy the hypotheses of the theorem. Also, sharply 2-transitive groups have been completely classified by Zassenhaus [44]. This theorem is one of a number of results of a similar nature. Zassenhaus groups are eas...

We prove that if f(x) = ∑n−1 k=0 akx k is a polynomial with no cyclotomic factors whose coefficients satisfy ak ≡ 1 mod 2 for 0 ≤ k < n, then Mahler’s measure of f satisfies log M(f) ≥ log 5 4 ( 1 − 1 n ) . This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n−1, and at least one noncyclotomic fact...

In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomol...

A simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field. For a polynomial f(x, y) over F of total degree n , our algorithm takes at most 4.89, 2 , n log n log q operations in F to factor f(x , y) completely. This improves a probabilistic factorization algorithm of von zur Gathen and Kaltofen, which takes 0(n log n log q) ...

We determine character tables for twisted fractional linear groups that form the “other” family in Zassenhaus’ classification of finite sharply 3-transitive groups.