A system of polynomial ordinary differential equations (odes) is specified via a vector of multivariate polynomials, or vector field, F. A safety assertion $\psi ⟶[F]\varphi$ means that the trajectory of the system will lie in a subset ϕ (the postcondition) of the state-space, whenever the initial state belongs to a subset ψ (the precondition). We consider the case when ϕ and ψ 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 ψ. 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 ψ, finds the largest subset of polynomials in P implied by ψ (relativized strongest postcondition). Under certain assumptions on ϕ, this algorithm can also be used to find the largest algebraic invariant included in ϕ and the weakest algebraic precondition for ϕ. Applications to continuous semialgebraic systems are also considered. The effectiveness of the proposed algorithm is demonstrated on several case studies from the literature.

多项式常微分方程（ode s）系统是通过多元多项式的向量或向量场F指定的。安全声明$\psi ⟶[F]\varphi$装置，该系统的轨迹将位于在一个子集φ的状态空间，每当初始状态属于子集的（后置条件）ψ（前提）。我们考虑ϕ和ψ为代数形式，即多项式为零集的情况。特别是，指定后置条件的多项式可以视为ψ所隐含的系统守恒律。例如，在混合系统中，检查代数安全性断言的有效性是一个基本问题。我们考虑此问题的广义形式，并提供一种算法，该算法在给定用户指定的多项式集P和代数前提ψ的情况下，找到ψ所隐含的P中多项式的最大子集（相对最强后置条件）。在一定的假设φ，该算法也可用于查找包括在最大代数不变φ和最薄弱的代数前提φ。还考虑了在连续半代数系统中的应用。文献中的一些案例研究证明了该算法的有效性。