Prime and composite Laurent polynomials

Pakovich, Fedor

doi:10.1016/j.bulsci.2009.06.003

Cited by 43 publications

(92 citation statements)

References 23 publications

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“…Nevertheless, it turns out that all reducible curves h P,Q (x, y) for which C(P ) ∩ C(Q) contains two elements can be described explicitly. In order to obtain such a description (and another proof of the first part of Proposition 3.1) we will use the following result which is due to Fried (see [20], Proposition 2, [21], Lemma 4.3, or [16], Theorem 3.5).…”

Section: ) P Is a Polynomial And Q Is A Rational Function With No Mumentioning

confidence: 99%

“…In the second section we recall a construction from [16] which permits to describe irreducible components of (5) and (6) and to calculate their genuses. In the third section we give several conditions implying the irreducibility of curves (5) and (6).…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 2.1 (( [16])) The Riemann surfaces R j , 1 ≤ j ≤ o(P, Q), are in a one-to-one correspondence with irreducible components of the curve h P,Q (x, y). Furthermore, each R j is a desingularization of the corresponding component.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic curvesP(x) −Q(y) = 0 and functional equations

Pakovich¹

2011

Complex Variables and Elliptic Equations

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In this paper we give several conditions implying the irreducibility of the algebraic curve P (x)−Q(y) = 0, where P, Q are rational functions. We also apply the results obtained to the functional equations P (f ) = Q(g) and P (f ) = cP (g), where c ∈ C. For example, we show that for a generic pair of rational functions P, Q the first equation has no non-constant solutions f, g meromorphic on C whenever (deg P − 1)(deg Q − 1) ≥ 2.

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Section: ) P Is a Polynomial And Q Is A Rational Function With No Mumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic curvesP(x) −Q(y) = 0 and functional equations

Pakovich¹

2011

Complex Variables and Elliptic Equations

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“…Generalizations of this problem include decomposition of multivariate polynomials [4,22], rational functions [30], algebraic functions [11] and Laurent polynomials [15,29]. The relationship between polynomial composition and polynomial systems has also been studied [7,8,9].…”

Section: The Problem Polynomial Decompositionmentioning

confidence: 99%

Functional Decomposition of Symbolic Polynomials

Watt

2008

2008 International Conference on Computational Sciences and Its Applications

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Earlier work has presented algorithms to factor and compute GCDs of symbolic Laurent polynomials, that is multivariate polynomials whose exponents are themselves integer-valued polynomials. This article extends the notion of univariate polynomial decomposition to symbolic polynomials and presents an algorithm to compute these decompositions. For example, the symbolic polynomial2 + 1 and h(X) = X n 2 /2+n/2 − X n 2 /2−n/2 .

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“…Up until now, analogues of Ritt's theory for rational functions have only been constructed for some special classes of these functions, for instance, for Laurent polynomials (Pakovich, 2009). In this note we describe a certain class of double decompositions (1) with rational functions R j (x) of degree greater than 1.…”

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Rational functions admitting double decompositions

Богатырев¹

2013

Trans. Moscow Math. Soc.

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Abstract. Ritt (1922) studied the structure of the set of complex polynomials with respect to composition. A polynomial P (x) is said to be indecomposable if it can be represented as P = P 1 •P 2 only if either P 1 or P 2 is a linear function. A decomposition P = P 1 •P 2 •. . .•P r is said to be maximal if all the P j are indecomposable polynomials that are not linear. Ritt proved that any two maximal decompositions of the same polynomial have the same length r, the same (unordered) set {deg(P j )} of the degrees of the composition factors, and can be connected by a finite chain of transformations each of which consists in replacing the left-hand side of the double decompositionby its right-hand side. Solutions of this functional equation are indecomposable polynomials of degree greater than 1, and Ritt listed all of them explicitly.Up until now, analogues of Ritt's theory for rational functions have only been constructed for some special classes of these functions, for instance, for Laurent polynomials (Pakovich, 2009). In this note we describe a certain class of double decompositions (1) with rational functions R j (x) of degree greater than 1. In essence, the rational functions described below were discovered by Zolotarëv as solutions of a certain optimization problem (1932). However, the double decomposition property for these functions remained little known because they had an awkward parametric representation. Below we give a representation for Zolotarëv fractions (possibly new), which resembles the well-known representation for Chebyshëv polynomials. These, by the way, are a special limit case of Zolotarëv fractions. § 1. Zolotarëv fractions and their composition property A purely imaginary parameter τ ∈ iR + defines the rectangle Π(τ ) of size 2 × |τ |:The conformal map x τ (u) taking this rectangle to the upper half-plane and fixing the three points u = ±1, 0 has the following simple form:in terms of the elliptic sine sn and the complete elliptic integral K. It is easy to deduce from the reflection principle for a conformal map that the parametric representation R(u) := x τ (u); x(u) := x nτ (u), u ∈ C, n ∈ N, defines a rational function R of degree n in the argument x depending parametrically on τ :Z n (x|τ ) := R(u(x)) = x τ • x −1 nτ .

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Prime and composite Laurent polynomials

Cited by 43 publications

References 23 publications

Algebraic curvesP(x) −Q(y) = 0 and functional equations

Algebraic curvesP(x) −Q(y) = 0 and functional equations

Functional Decomposition of Symbolic Polynomials

Rational functions admitting double decompositions

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