We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion. © 2014 Springer-Verlag.
CITATION STYLE
Ghorbal, K., & Platzer, A. (2014). Characterizing algebraic invariants by differential radical invariants. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8413 LNCS, pp. 279–294). Springer Verlag. https://doi.org/10.1007/978-3-642-54862-8_19
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