Characterizing Algebraic Invariants by Differential Radical Invariants

Abstract

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.

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

“…The proof is in the companion report [12]. When the evolution domain constraints are dropped (H = True) and r = 1 (one equation), one recovers exactly the statement of [11,Theorem 2] which characterizes invariance of atomic equations. Intuitively, Theorem 2 says that on the invariant algebraic set, all higher-order Lie derivatives of each polynomial h i must vanish.…”

“…Using Theorem 2, the differential radical invariant proof rule DRI [11] generalizes to conjunctions of equations with evolution domain constraints as follows:…”

“…For similar techniques used to automatically generate invariant algebraic sets we refer the reader to the discussion in [11].…”

Decoupling Abstractions of Non-linear Ordinary Differential Equations

“…In [7], the authors extended the work of Matringe et al to generate real algebraic invariants of polynomial ODEs, giving a search procedure for the most general class of invariant sets that can be expressed using polynomial equations. The same procedure can be used to generate Darboux polynomials over the reals or over the complexes only by changing the underlying computational field.…”

“…This relationship will enable us to exploit the efficient symbolic generation methods reported in [15,7]. We outline a procedure for constructing polynomials p such that L f (p) = G(p), where G ∈ R[X], from a list of automatically generated Darboux polynomials (up to some given degree).…”