Triangular derivations related to problems on affine 𝑛-space

Essen, Arno van den; Rossum, Peter van

doi:10.1090/s0002-9939-01-06212-8

Cited by 8 publications

(6 citation statements)

References 18 publications

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“…Added in Proof. The author would like to express his thanks to Professor Michiel de Bondt, who informed the candidate counterexample [5] to the author. The author wonders why Conjecture E has been left unsettled for more than ten years.…”

Section: The Main Resultsmentioning

confidence: 99%

“…If Conjecture E is affirmative, it gives a negative answer to the Cancellation Problem and to the Linearization Problem, and shows that Shastri's embedding is indeed a counterexample to the Embedding Problem and the A n -Fibration Problem (n ≥ 3) over a line A 1 C . However, by The Main Result 2, then they are also still open by [5].…”

Section: Note Thatmentioning

confidence: 94%

“…which is easily seen to be a locally nilpotent derivation on A and has a slice s [5] but A D ∼ =C C [4] . This conjecture is equivalent to the following conjecture.…”

Section: Notationsmentioning

confidence: 99%

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Note on the candidate counter-example in the cancellation problem for affine spaces posed by Arno Van den Essen

Oda

2012

Preprint

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We have proved the following Problem:Let R be a C-affine domain, let T be an element in R \ C and let i : n] . This result leads to the negative solution of the candidate counter-example (Conjecture E) of V.Arno den Essen :Let A := C[t, u, x, y, z] denote a polynomial ring, and let f (u) := u 3 − 3u, g(u) := u 4 − 4u 2 and h(u) := u 5 − 10u be the polynomials in C 4] . Consequently our result in this short paper guarantees the conjectures : the Cancellation Problem for affine spaces", "the Linearization Problem", "the Embedding Problem" and "the affine A n -Fibration Problem" to be still open.

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Section: The Main Resultsmentioning

confidence: 99%

Section: Note Thatmentioning

confidence: 94%

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Note on the candidate counter-example in the cancellation problem for affine spaces posed by Arno Van den Essen

Oda

2012

Preprint

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“…. ., X n ] be a polynomial ring over a field K. A natural problem in commutative algebra and algebraic geometry is to understand the group GA n (K) of automorphisms of K[X] preserving K. There are various long-standing open problems and conjectures in affine algebraic geometry concerning polynomial rings and their automorphisms (see [10], [11] and [15] for more details). Below we mention a few of the most famous ones.…”

Section: Introductionmentioning

confidence: 99%

Linearized polynomial maps over finite fields

Berson

2014

Journal of Algebra

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We consider polynomial maps described by so-called (multivariate) linearized polynomials. These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F q automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show that the linearized polynomial maps over F q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F q . Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible linearized polynomial maps correspond to invertible matrices.This alternate description of the linearized polynomial automorphism subgroup leads to the solution of many famous conjectures (most notably, the Jacobian Conjecture) for this kind of polynomials and polynomial maps.

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“…One of the most fundamental questions in studying problems on affine n-space is the following : given a polynomial F in n variables over a field K, how can one decide if F is a coordinate ? Related problems are the Abhyankar-Sathaye Conjecture, the Cancellation Problem, the Jacobian Conjecture and Linearization Conjectures (see [10], [9] and [7] for more details). In case n = 2 this problem was solved by Ch §dzynski and Krasinski in [3] and by the second author in [8]; in fact, in the last paper also an algorithm was given to compute a mate of F. However the case n > 3 remains open.…”

Section: Introductionmentioning

confidence: 99%