The defect formula

In this paper we present a characterization for the defect of a simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let $\text{kh}$ be a henselization of a valued field $K$. Another relevant result proved in this paper is that for every valuation $\text{muh}$ on $\text{khx}$, with restriction $\mu$ on $\text{kx}$, the corresponding map ${\mathcal{G}}_{\mu }\text{hk}{\mathcal{G}}_{\text{muh}}$ of graded algebras is an isomorphism.