  A system of polynomial ordinary differential equations (ODEs) is specified
via a vector of multivariate polynomials, or vector field, $F$. A safety
assertion $\psi\rightarrow[F]\phi$ means that the trajectory of the system will
lie in a subset $\phi$ (the postcondition) of the state-space, whenever the
initial state belongs to a subset $\psi$ (the precondition). We consider the
case when $\phi$ and $\psi$ are algebraic varieties, that is, zero sets of
polynomials. In particular, polynomials specifying the postcondition can be
seen as a system's conservation laws implied by $\psi$. Checking the validity
of algebraic safety assertions is a fundamental problem in, for instance,
hybrid systems. We consider a generalized version of this problem, and offer an
algorithm that, given a user specified polynomial set $P$ and an algebraic
precondition $\psi$, finds the largest subset of polynomials in $P$ implied by
$\psi$ (relativized strongest postcondition). Under certain assumptions on
$\phi$, this algorithm can also be used to find the largest algebraic invariant
included in $\phi$ and the weakest algebraic precondition for $\phi$.
Applications to continuous semialgebraic systems are also considered. The
effectiveness of the proposed algorithm is demonstrated on several case studies
from the literature.
