Very good and very bad field generators

Cassou-Noguès, Pierrette; Daigle, Daniel

doi:10.1215/21562261-2848160

Cited by 3 publications

(8 citation statements)

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“…The above theorem is one of the main results of [CND14a]. Its corollary (below) has interesting consequences in the classification of field generators (for instance, 3.9(a) is needed in the proof of 4.1(b)).…”

Section: Definition Letmentioning

confidence: 87%

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Field generators in two variables and birational endomorphisms of 𝔸²

Cassou-Noguès

Daigle

2015

J. Algebra Appl.

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This article is a survey of two subjects: the first part is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. Note that the part on field generators is more than a survey, since it contains a considerable amount of new material.This article is a survey of two subjects: the first part of the paper is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. The authors of the present article were introduced to these questions by Peter Russell, who participated in Abhyankar's seminar and who made early contributions to both problems.As explained in Section 1, the two subjects are entangled one into the other. It is therefore natural to present them together in a survey. Note that Part I is more than a survey, since it contains a considerable amount of new material (see Section 1); and that Part II is less than a survey, since it restricts itself to certain particular aspects of the subject under consideration (see Section 6). ConventionsThe symbol "⊂" means strict inclusion of sets, "\" means set difference, and 0 ∈ N.If R is a subring of a ring S, the notation S = R [n] means that S is R-isomorphic to a polynomial algebra in n variables over R. If L/K is a field extension, L = K (n) means that L is a purely transcendental extension of K, of transcendence degree n. We write Frac R for the field of fractions of a domain R.If k is a field and A = k [2] (i.e., A is a polynomial ring in two variables over k) then a coordinate system of A is an ordered pair (X, Y ) ∈ A × A satisfying A = k[X, Y ]. We define C(A) to be the set of coordinate systems of A. A variable of A is an element X ∈ A satisfying A = k[X, Y ] for some Y .

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Section: Definition Letmentioning

confidence: 87%

“…Our next objective is to study how ∆(F, A) and Γ(F, A) behave under a birational extension of A. This is accomplished by 2.13 and 2.14, which are respectively results 2.9 and 3.11 of [CND14a]. See the introduction for the notation "A B".…”

Section: Notation Consider Morphismsmentioning

confidence: 99%

Field generators in two variables and birational endomorphisms of 𝔸²

Cassou-Noguès

Daigle

2015

J. Algebra Appl.

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“…Refer to [6] for details on these notions (see also [20,3]). "Good" and "very good" field generators appear in Lemma 3.2, Theorem 3.3 and Theorem 6.7, below.…”

Section: Corollary Consider Morphismsmentioning

confidence: 99%

“…The second part of Section 6 presents two results on lean factorizations, namely, we show that every dominant morphism A 2 → A 2 admits a lean factorization (Theorem 6.5), and we determine which dominant morphisms A 2 → A 1 admit a lean factorization (Theorem 6.7). The insight provided by Theorem 6.7 allows us, in [6] and [5], to make some progress in the open problem of classifying rational polynomials (let us say, briefly, that a rational polynomial is a dominant morphism A 2 → A 1 whose general fibers are rational curves). The relation between lean factorizations and the classification of rational polynomials is explained in Section 1 of [5].…”

Section: Introductionmentioning

confidence: 99%

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Sets of polynomial rings in two variables and factorizations of polynomial morphisms

Cassou-Noguès

Daigle

2016

Journal of Algebra

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Let k be a field. We study infinite strictly descending sequences A 0 ⊃ A 1 ⊃ · · · of rings where each A i is a polynomial ring in two variables over k, the aim being to describe those sequences satisfying ∞ i=0 A i = k. We give a complete answer in characteristic zero, and partial results in arbitrary characteristic. We apply those results to the study of dominant morphisms A 2 → A n and their factorizations, where n ∈ {1, 2} and A n is the affine n-space over k.

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