Abstract not found

We show that for any even positive integer δ there exist polynomials x and y with integer coefficients such that deg(x) = 2δ, deg(y) = 3δ and deg(x 3 − y 2) = δ + 5. Hall’s conjecture asserts that for any ε> 0, there exists a constant c(ε)> 0 such that if x and y are positive integers satisfying x3 − y2 ̸ = 0, then |x3 − y2 |> c(ε)x1/2−ε. It is known that Hall’s conjecture follows from the abc-conjecture. For a stronger version of Hall’s conjecture which is equivalent to the abc-conjecture see [3, Ch. 12.5]. Originally, Hall [8] conjectured that there is C> 0 such that |x3 − y2 | ≥ C √ x for positive integers x, y with x3 − y2 ̸ = 0, but this formulation is unlikely to be true. Danilov [4] proved that 0 < |x3 −y 2 | < 0.97 √ x has infinitely many solutions in positive integers x, y; here 0.97 comes from 54 √ 5/125. For examples with “very small ” quotients |x3 − y2 | / √ x, up to 0.021, see [7] and [9]. It is well known that for non-constant complex polynomials x and y, such that x3 ̸ = y2, we have deg(x3 − y2) / deg(x)> 1/2. More precisely, Davenport [6] proved that for such polynomials the inequality deg(x 3 − y 2) ≥ 1 2 deg(x) + 1 (1) holds. This statement also follows from Stothers-Mason’s abc theorem for polynomials (see, e.g., [10, Ch. 4.7]). Zannier [12] proved that for any positive integer δ there exist complex polynomials x and y such that deg(x) = 2δ, deg(y) = 3δ and x, y satisfy the equality in Davenport’s bound (1). In his previous paper [11], he related the existence of such examples with coverings of the Riemann sphere, unramified except above 0, 1 and ∞.