In this paper, local stability and local stabilization problems are considered for discrete-time nonlinear systems represented by Takagi-Sugeno (T-S) fuzzy systems. Improved methods to assess the local stability, design locally stabilizing control laws, and estimate the domain of attraction are developed in terms of single-parameter minimization problems subject to linear matrix inequality (LMI) constraints. The improvement is achieved by applying a convergent LMI relaxation technique to prove positivity of homogeneous polynomial parameter-dependent matrices of arbitrary degree with variables in the simplex. We also take advantage of the recently developed parameter variation modeling technique to deal with the case that the variation rate of the membership functions of the T-S fuzzy systems is bounded and less than one. Finally, several illustrative examples demonstrate the validity and efficiency of the proposed method.