2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149832This paper, the last in a series of three, studies vector bundles on an elliptic surface whose determinant has odd intersection number with a general fiber and uses this study to calculate certain coefficients of Donaldson polynomials.68 pages, AMS-TeXAlgebraic GeometryVector bundles and $SO(3)$ invariants for elliptic surfaces III: The case of odd fiber degreetext