The research presented in this thesis lies in the field of applied algebraic geometry. The main focus is on developing algebraic methods for studying the local dynamics of steady states in systems of polynomial differential equations with parametric coefficients. Inparticular, in systems arising from chemical reaction network models of biochemical processes. In this thesis we present the contributions made in two topics: effective methods for detecting bistability and Hopf bifurcations in a chemical reaction network, and detection of Absolute Concentration Robustness in chemical reaction networks. This document has three parts. In the first part we present the background required for our contributions. It contains the basic definitions of chemical reaction network theory, a small survey on methods for assessing the presence of multistationarity, and background in polytopes and the Newton polytope of a polynomial. The second part contains our contributions regarding bistability and Hopf bifurcations. We outline an algorithm to guarantee or preclude bistability in chemical reaction networks satisfying certain conditions. This algorithm is symbolic and can be used to find parameter regions for bistabillity. Additionally, we use the Newton polytope and its outer normal fan to effectively compute a set of parameters and a steady state where a Hopf bifurcation arises in a model of ERK regulation and in a MAPK cascade. Finally, in the third part, we present our contributions related with Absolute Concentration Robustness (ACR). In particular, we present a graphic method for finding ACR in small networks and the n, we study whether ACR is preserved under structural modifications of the network.