  For polynomial ODE models, we introduce and discuss the concepts of exact and
approximate conservation laws, which are the first integrals of the full and
truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast
dynamics. We define compatibility classes as subsets of the state space,
obtained by equating the conservation laws to constants. A set of conservation
laws is complete when the corresponding compatibility classes contain a finite
number of steady states. Complete sets of conservation laws can be used for
model order reduction and for studying the multistationarity of the model. We
provide algorithmic methods for computing linear, monomial, and polynomial
conservation laws of polynomial ODE models and for testing their completeness.
The resulting conservation laws and their completeness are either independent
or dependent on the parameters. In the latter case, we provide parametric case
distinctions. In particular, we propose a new method to compute polynomial
conservation laws by comprehensive Gr\"obner systems and syzygies.
  Keywords: First integrals, chemical reaction networks, polynomial
conservation laws, syzygies, comprehensive Gr\"obner systems.
