Kyoto Journal of Mathematics

Very good and very bad field generators

Pierrette Cassou-Noguès and Daniel Daigle

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Let k be a field. A field generator is a polynomial Fk[X,Y] satisfying k(F,G)=k(X,Y) for some Gk(X,Y). If G can be chosen in k[X,Y], we call F a good field generator; otherwise, F is a bad field generator. These notions were first studied by Abhyankar, Jan, and Russell in the 1970s. The present paper introduces and studies the notions of “very good” and “very bad” field generators. We give theoretical results as well as new examples of bad and very bad field generators.

Article information

Kyoto J. Math., Volume 55, Number 1 (2015), 187-218.

First available in Project Euclid: 13 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14H50: Plane and space curves

Affine plane birational morphism plane curve rational polynomial field generator dicritical


Cassou-Noguès, Pierrette; Daigle, Daniel. Very good and very bad field generators. Kyoto J. Math. 55 (2015), no. 1, 187--218. doi:10.1215/21562261-2848160.

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