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Polynomial Rings

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Algebraic Theory of Locally Nilpotent Derivations

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 136.3))

Abstract

This chapter investigates locally nilpotent derivations in the case B is a polynomial ring in a finite number of variables over a field k of characteristic zero. Equivalently, we are interested in the algebraic actions of \(\mathbb{G}_{a}\) on \(\mathbb{A}_{k}^{n}\).

Locally nilpotent derivations are useful if rather elusive objects. Though we do not have them at all on “majority” of rings, when we have them, they are rather hard to find and it is even harder to find all of them or to give any qualitative statements. We do not know much even for polynomial rings.

Leonid Makar-Limanov, Introduction to [ 282 ]

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Notes

  1. 1.

    Ernest Jean Philippe Fauque de Jonquières (1820–1901) was a career officer in the French navy, achieving the rank of vice-admiral in 1879. He learned advanced mathematics by reading works of Poncelet, Chasles, and other geometers. In 1859, he introduced the planar transformations \((x,y) \rightarrow \left (x, \frac{a(x)y+b(x)} {c(x)y+d(x)} \right )\), where ad − bc ≠ 0. These were later studied by Cremona.

  2. 2.

    Van den Essen gives a more exclusive definition of a nice derivation. See [142], 7.3.12.

  3. 3.

    Some authors use DF to denote the jacobian matrix of F, but we prefer to reserve D for derivations.

  4. 4.

    “The functions Φ(x), constructed for the arguments x + λξ, are independent of λ.”

  5. 5.

    “If such an entire function Φ is a product of two entire functions Φ = ϕ(x)ψ(x), then so also are the factors themselves functions Φ.”

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Authors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, Michigan, USA

    Gene Freudenburg

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  1. Gene Freudenburg
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Freudenburg, G. (2017). Polynomial Rings. In: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences, vol 136.3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55350-3_3

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