Trinomials with given roots

Abstract

Riffaut [18] conjectured that a singular modulus of degree h ≥ 3 cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.

Binary polynomial power sums vanishing at roots of unity

“…Using this, we deduce from (4.10) the inequality n 2/3 ≤ 10 15 d 5 (log(3d)) 3 . A quick calculation shows that this is incompatible with (4.1).…”

“…In the set-up of Theorem 1.2, assume that, for a given m, the set of positive integers n with the property "the polynomial u n (x) is not identically 0 but vanishes at an mth root of unity" is not empty. Then the smallest n in this set satisfies n ≤ m(log m) 3 (X + log D).…”

“…More precisely, either there exists n in this set satisfying n ≤ 2m, or every n in this set satisfies n ≤ m(log m) 3 (X + log D).…”

On the prime factors of the iterates of the Ramanujan τ–function

“…We need a lower bound for |η a+1 − 1|. Corollary 4.2 in [6] shows that if we write D and h(η) for the degree and logarithmic height of η, respectively, then the inequality…”

“…. , τ ( k/2 ) (n)} (6) there is an even number, then with m := τ ( k/2 ) (n), we have that m is even and Proposition 1 shows that P (τ (k) (n)) = P (τ (k− k/2 ) (m)) ≥ 3 k− k/2 −1 + 2 ≥ 3 k/2−1 + 2 > log(k/2) for k > 10, and therefore P (Orb τ (n, k)) ≥ P (m) > log(k/2). Thus, we may assume that all numbers in list L given at (6) are odd.…”