On polynomial Solutions of a differential equation

Magnus, Arne

doi:10.7146/math.scand.a-10443

Cited by 20 publications

(10 citation statements)

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“…Denote w := 15(x−y) 14 . Then Jac(p, q) = (1+w Condition (i) is not satisfied, so none of Theorems 2.4 and 2.6 is applicable.…”

Section: Examplesmentioning

confidence: 99%

A variation on Magnus' theorem and its generalizations

Moskowicz

2018

Preprint

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Let k be a field of characteristic zero, and let f : k[x, y] → k[x, y], f : (x, y) → (p, q), be a k-algebra endomorphism having an invertible Jacobian.WriteDenote the set of prime numbers by P . Under two mild conditions, we prove that, if gcd(gcd(n, deg x (an)), gcd(r, deg x (cr))) ∈ {1, 8} ∪ P ∪ 2P, then f is an automorphism of k[x, y].Removing (at least one of) the two mild conditions, we present two additional results. One of the additional results implies that the known form of a counterexample (P, Q) to the two-dimensional Jacobian Conjecture, l 1,1 (P ) = ǫx αµ y βµ , l 1,1 (Q) = δx αν y βν , where ǫ, δ ∈ k × , 1 < α < β, d := gcd(α, β) > 1, 1 < ν < µ, gcd(µ, ν) = 1, actually satisfies d > 2.

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“…Denote w := 15(x−y) 14 . Then Jac(p, q) = (1+w Condition (i) is not satisfied, so none of Theorems 2.4 and 2.6 is applicable.…”

Section: Examplesmentioning

confidence: 99%

A variation on Magnus' theorem and its generalizations

Moskowicz

2018

Preprint

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“…6) So, it is useful to have a sure flatness criterion for unramified rings morphism. Besides a criterion deduced from Nagata's flatness criterion and another one deduced from the jacobian criterion of smoothnes in [17] (see [2],1.1 (15) and (16) for more detail), the most interesting of this kind of criterion is the following one given by A. Grothendieck in [19],Expos I, Cor. 9.11.…”

Section: )mentioning

confidence: 99%

On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic

Adjamagbo

1995

Automorphisms of Affine Spaces

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The main theorem (2.2) consists in two characterizations of isomorphisms of factorial domains in terms of prime or primary rings elements, and unramified, flat or weakly injective affine schemes morphisms. In order to apply this theorem to the famous Jacobian Conjecture, we first introduce its different versions in any characteristic (3.1), and give two reformulations of some these versions in terms of domains of positive characteristic (3.8) and finite prime fields (3.9). Finally, we deduce from the main theorem an original reformulation of the any characteristic version of the Jacobian Conjecture in terms of prime or primary rings elements (3.11).

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“…In [23, Theorem 1], Magnus produced a formula which inspired much of the work for this paper. His formula was published almost 70 years ago but has not been used in almost any paper but [24]. Even in [24], only small piece of information from the formula was utilized.…”

Section: Introductionmentioning

confidence: 99%

“…His formula was published almost 70 years ago but has not been used in almost any paper but [24]. Even in [24], only small piece of information from the formula was utilized. The first main result in our paper is a more general version of Magnus' formula, as given in Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

Hurst¹,

Lee²,

Li³

et al. 2022

Preprint

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Let K be an algebraically closed field of characteristic 0. When the Jacobian (∂f /∂x)(∂g/∂y) − (∂g/∂x)(∂f /∂y) is a constant for f, g ∈ K[x, y], Magnus' formula from [23] describes the relations between the homogeneous degree pieces f i 's and g i 's. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.

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On polynomial Solutions of a differential equation

Cited by 20 publications

References 1 publication

A variation on Magnus' theorem and its generalizations

A variation on Magnus' theorem and its generalizations

On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

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