Fixed numerators in the Forgács–Tran family: bounded defect and angular zero discrepancy

Forgács and Tran proved that the polynomial sequence generated by the reciprocal of \(Q(t)+zt^r\), with \(Q\) having only positive real zeros, is eventually real-rooted with all zeros in an explicit positive interval. Their Problem 14 asks what survives when a proper numerator, equivalently arbitrary fixed polynomial initial conditions, is introduced. No bound can be uniform over all numerators, but for each fixed numerator the number of zeros off the positive ray is bounded independently of the coefficient index, and eventually the same is true outside the Forgács–Tran interval. Restricting the numerator to the denominator curve produces a canonical univariate weight that carries its asymptotic effect. The eventual defect is bounded linearly in the weight's degree, and the normalized zero distribution is independent of the numerator; the same weight governs the finer local zero geometry.