Linearization of polynomial flows and spectra of derivations

Coomes, Brian A.; Zurkowski, V. D.

doi:10.1007/bf01049488

Cited by 9 publications

(7 citation statements)

References 11 publications

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“…This result in turn was improved in a paper of Coomes and Zurkowski [10]; they showed (v) A polynomial flow of any non-linear «-dimensional system is the solution of a linear ODE, of some sufficient high order.…”

Section: Introductionmentioning

confidence: 94%

“…To illustrate that statement we discussed several examples. Using properties of locally finite and locally nilpotent derivations we gave new very simple proofs of the fact that the Lorenz equations and the Maxwell-Bloch equations do not have a polynomial flow (results first obtained by Coomes in [7,8] and Coomes and Zurkowski in [10]). Also a very short proof of the Bass-Meisters classification theorem of two dimensional polynomial flows (cf.…”

Section: Introductionmentioning

confidence: 99%

“…In § 1 we recall some well-known facts concerning derivations and give a useful formula describing the solutions of a system of ordinary differential equations of the form y = P(y). In §2 we apply the results of section one to give simplified proofs of some results of Coomes and Zurkowski [9,10] concerning polynomial flows. At the end ofthat section, inspired by the papers of Meisters [18], Olech and Meisters [19], and Adjamagbo [1], we give a new inversion formula for polynomial automorphisms in terms of a locally nilpotent derivation.…”

Section: Introductionmentioning

confidence: 99%

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Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms and 𝒢ₐ-actions. II

Essen¹

1994

Proc. Amer. Math. Soc.

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Abstract. This paper describes several applications of locally finite and locally nilpotent derivations; we recover the main results from the theory of polynomial flows, give a new inversion formula for polynomial automorphisms, reformulate the Eulerian conjecture for ordinary systems of differential equations in terms of locally nilpotent derivations, and give an algorithm to compute the invariant ring of a ^,-action on affine /i-space (if this ring is finitely generated).

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms and 𝒢ₐ-actions. II

Essen¹

1994

Proc. Amer. Math. Soc.

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“…The following result is due to H. Bass, G. Meisters [2] and B. Coomes, V. Zurkowski [4]. Another its proof is given in [19] (Theorem 9.7.3).…”

Section: Preliminariesmentioning

confidence: 99%

Divergence-free polynomial derivations

Nowicki¹

2017

Analytic and Algebraic Geometry 2

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Abstract. In this paper we present some new and old properties of divergences and divergence-free derivations.Throughout the paper all rings are commutative with unity. Let k be a ring and let d be a k-derivation of the polynomial ringThe derivation d is said to be divergence-free if d = 0. PreliminariesLet k be a ring, and let R be a k-algebra. A k-linear mapping d : R → R is said to be a k-derivation of R iffor all a, b ∈ R. We denote by Der k (R) the set of all k-derivations of R.Thus, the set Der k (R) is an R-module which is also a Lie algebra.We denote by R d the kernel of d, that is,This set is a subring of R, called the ring of constants of R (with respect to d). If R is a field, then R d is a subfield of R.

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“…Coomes and Zurkowski [8] show that 0 is a polynomial flow if and only if T(D) = C[n]. (See [3,5,6,7,10,11,12,13,14,15,17] for other results about polynomial flows.)…”

Section: Introductionmentioning

confidence: 99%