Prime and composite polynomials

Dorey, F; Whaples, G.

doi:10.1016/0021-8693(74)90023-4

Cited by 57 publications

(53 citation statements)

References 8 publications

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“…This is also false for rational functions, see [4] and [1]. We look for bounds for the degree of the extension in which we need to take the coefficients if a rational function with coefficients in Q has a decomposition in a larger field.…”

Section: Remarkmentioning

confidence: 98%

Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem

Gutiérrez

Sevilla

2006

Journal of Algebra

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The classical Ritt's theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to the First Ritt Theorem, which states the equality of length of decomposition chains of a polynomial, in the case of rational functions. Namely, we provide an explicit example of a rational function with coefficients in Q and two decompositions of different length.Another aspect is the use of some techniques that could allow for other counterexamples, namely, relating groups and decompositions and using the fact that the alternating group A 4 has two subgroup chains of different lengths; and we provide more information about the generalizations of another property of polynomial decomposition: the stability of the base field. We also present an algorithm for computing the fixing group of a rational function providing the complexity over the rational number field.

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Section: Remarkmentioning

confidence: 98%

Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem

Gutiérrez

Sevilla

2006

Journal of Algebra

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“…(4) mean that α = p 1 · · · p r−1 and β = q 1 · · · q s−1 . (5) Recall that we have the equality |c| = |p r ||q s |. In combination with (3), i.e.…”

Section: Generalizations Of Ritt's Theorems For Reduction Monoidsmentioning

confidence: 98%