Displaying similar documents to “A Note on Elementary Derivations”

A Survey of Counterexamples to Hilbert's Fourteenth Problem

Freudenburg, Gene (2001)

Serdica Mathematical Journal

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We survey counterexamples to Hilbert’s Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 important open questions.

On the ring of constants for derivations of power series rings in two variables

Leonid Makar-Limanov, Andrzej Nowicki (2001)

Colloquium Mathematicae

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Let k[[x,y]] be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k[[x,y]]. We prove that if Ker(d) ≠ k then Ker(d) = Ker(δ), where δ is a jacobian derivation of k[[x,y]]. Moreover, Ker(d) is of the form k[[h]] for some h ∈ k[[x,y]].

Some results on the kernels of higher derivations on k[x,y] and k(x,y)

Norihiro Wada (2011)

Colloquium Mathematicae

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Let k be a field and k[x,y] the polynomial ring in two variables over k. Let D be a higher k-derivation on k[x,y] and D̅ the extension of D on k(x,y). We prove that if the kernel of D is not equal to k, then the kernel of D̅ is equal to the quotient field of the kernel of D.