Elimination Theory in Differential and Difference Algebra

Differential Elimination and Algebraic Invariants of Polynomial Dynamical Systems

“…This makes the theory a natural choice for exploring algebraic invariants of polynomial dynamical systems. In particular, differential elimination [27] is the algorithmic engine that powers our main results, Theorem 86 in Section 3 and Theorem 98 in Section 4.…”

“…In recent decades, researchers in differential algebra [26] have extended classical algebraic elimination methods to polynomial differential equations (Section 2.4). Broadly speaking, differential elimination [27] provides tools for finding all relations implied by a polynomial differential system, regardless of the form. The elimination procedures we develop in Sections 3 and 4 are intrinsically mathematical, but we show that the output corresponds directly to invariants of the input system, giving a novel, complementary view on systematically computing and checking ODE invariants.…”

Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory

“…Differential resultants were first defined for ordinary differential operators, as the natural generalization to a non commutative environment of the algebraic resultant of two univariate polynomials, see for instance [6]. A few years ago the theory of differential resultants was formalized for multivariate differential polynomials, as reviewed in two recent reports [22] and [20]. No attempt has been made so far to define a differential resultant for MODOs.…”