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FACTORIZATION OF MONOMORPHISMS OF A POLYNOMIAL ALGEBRA IN ONE VARIABLE

Published online by Cambridge University Press:  01 May 2008

V. V. BAVULA*
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK e-mail: [email protected]
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Abstract

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Let K[x] be a polynomial algebra in a variable x over a commutative -algebra K, and Γ′ the monoid of K-algebra monomorphisms of K[x] of the type σ: xx2x2 + . . . +λnxn, λiK, λn is a unit of K. It is proved that for each σ ∈ Γ′ there are only finitely many distinct decompositions σ = σ1. . .σs in Γ′. Moreover, each such decomposition is uniquely determined by the degrees of components: if σ = σ1. . . σs= τ1 . . . τs then σ11, λ. . ., σss if and only if deg(σ1)=deg(τ1), . . ., deg(σs)=deg(τs). Explicit formulae are given for the components σi via the coefficients λj and the degrees deg(σk) (as an application of the inversion formula for polynomial automorphisms in several variables from [1]). In general, for a polynomial there are no formulae (in radicals) for its divisors (elementary Galois theory). Surprisingly, one can write such formulae where instead of the product of polynomials one considers their composition (as polynomial functions).

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

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