3-Adic zeros of diagonal forms of odd degree

Consider a diagonal form of degree [Formula: see text] with [Formula: see text] and with integer coefficients. Let [Formula: see text] denote the smallest integer such that any such diagonal form has nontrivial 3-adic zeros whenever the number of variables is at least [Formula: see text]. We show that for odd k, if [Formula: see text], then [Formula: see text] is bounded above by [Formula: see text]. This result improves on previous bounds for diagonal forms of odd degree.