Asymptotic Expansion of the Equipoise Curve of a Polynomial Inequality

摘要

For any define and The two homogeneous polynomials and are comparable in the positive octant Recently the authors [2] studied the inequality tS_{mathbf{a}}(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 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 where the coefficients involve a type of generalised Catalan number, and the case where is a sequence in which each term is close to 1. A selection of inequalities implied by these results completes the paper.