Asymptotic Expansion of the Equipoise Curve of a Polynomial Inequality

For any a := (a1, a2, . . . , an) ∈ (R) , define ∆Pa(x, t) := (x+ a1t)(x+ a2t) · · · (x + ant) − x and Sa(x, y) := a1x + a2xy + · · · + any. The two homogeneous polynomials ∆Pa(x, t) and tSa(x, y) are comparable in the positive octant x, y, t ∈ R. Recently the authors [2] studied the inequality ∆Pa(x, t) > tSa(x, y) and its reverse and noted that the boundary between the corresponding regions in the positive octant is fully determined by the equipoise curve ∆Pa(x, 1) = Sa(x, y). In the present paper the asymptotic expansion of the equipoise curve is shown to exist, and is determined both recursively and explicitly. Several special cases are then examined in detail, including the general solution when n = 3, where the coefficients involve a type of generalised Catalan number, and the case where a = 1+ δ is a sequence in which each term is close to 1. A selection of inequalities implied by these results completes the paper.