Abstract
In this paper, we introduce an innovative and systematic technique to study delay differential equations via polynomials. First, we review an intrinsic relation between delay differential equations and polynomials. From this relation, we obtain long time behaviors of the solutions to delay differential equations via asymptotic analysis of the corresponding polynomials. Moreover, we derive asymptotic formulas and upper bounds for the intrinsic growth rate of delay differential equations, as well as a Gronwall-type inequality for delay differential inequalities.