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

Pakovich, Fedor

doi:10.1080/17476930903394838

Cited by 26 publications

(35 citation statements)

References 27 publications

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“…We refer the reader to Brockett (1983) for related work on Galois groups attached to linear feedback systems. Algebraic-geometric characterizations of decomposable rational functions in terms of root loci or associated Bezoutian curves have been obtained by Pakovich (2011).…”

Section: Notes and Referencesmentioning

confidence: 99%

The Mathematics of Networks of Linear Systems

Fuhrmann

Helmke

2015

Universitext

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Section: Notes and Referencesmentioning

confidence: 99%

The Mathematics of Networks of Linear Systems

Fuhrmann

Helmke

2015

Universitext

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“…Several authors have studied functional equations of type f (a) = g(b), where f, g are given complex polynomials, which are to be solved in non-constant meromorphic complex functions a, b, and in particular the case when g(x) = c f (x) for some c ∈ C. Recently, Pakovich [23] studied the case when one allows f and g to be rational functions. To the study of such questions of importance are questions about reducibility of the curve 2 , and f 1 , f 2 , as well as g 1 , g 2 , have no common roots (that is the curve obtained by equating to zero the numerator of f (x) = g(y)), and reducibility of the curve (…”

Section: Remark 311mentioning

confidence: 99%

“…In the present paper, these are the critical values at simple critical points. Pakovich [23] showed that if f and g have at most one common critical value, then the corresponding curve is irreducible. He further showed that the curve…”

Section: Remark 311mentioning

confidence: 99%

“…Namely, by Picard's theorem from complex analysis there exists a solution of the above functional equation if and only if the corresponding curve has an irreducible component of genus at most one. For a rational function f with complex coefficients, Pakovich [23] called s ∈ C a critical value of f if the set f −1 {s} contains less than deg f points. This is compatible with our definition of a critical value of a polynomial.…”

Section: Remark 311mentioning

confidence: 99%

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Diophantine equations in separated variables

Kreso

Tichy

2017

Period Math Hung

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We study Diophantine equations of type f (x) = g(y), where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials ( f n ) n are such that f n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if f (x) = g(h(x)) with g, h ∈ K [x] and deg g > 1, then either deg h ≤ 2, or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.

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“…Remark. As written, the proof of Theorem 1.2 relies on transcendental arguments via the results of [Pak11]. Nevertheless, I believe that it also holds in characteristic p > 0.…”

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confidence: 99%

Rational curves on elliptic surfaces

Ulmer

2016

J. Algebraic Geom.

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A. We prove that a very general elliptic surface E → P 1 over the complex numbers with a section and with geometric genus p g ≥ 2 contains no rational curves other than the section and components of singular fibers. Equivalently, if E/C(t) is a very general elliptic curve of height d ≥ 3 and if L is a finite extension of C(t) with L ∼ = C(u), then the Mordell-Weil group E(L) = 0. IFix a field k and consider elliptic curves E over K = k(t). When k is a finite field, we showed in [Ulm07] that there are often finite extensions L of K which are themselves rational function fields (i.e., L ∼ = k(u)) such that the rank of E(L) is as large as desired. Indeed, under a mild parity hypothesis on the conductor of E (which should hold roughly speaking in one half of all cases), we may take extensions of the form L = k(t 1/d ) with d varying through integers prime to p. More generally, for any elliptic curve E over K with j(E) ∈ k there is a finite extension of K of the form k ′ (u) with k ′ a finite extension of k such that E obtains unbounded rank in the layers of the tower k ′ (u 1/d ). Our aim in this note is to show that the situation is completely different when k is the field of complex numbers.From now on we take k = C. If E is an elliptic curve over C(t), the height of E is the smallest non-negative integer d such that E has a Weierstrass equationwhere a(t) and b(t) are polynomials of degree ≤ 4d and ≤ 6d respectively. Our results concern elliptic curves of height d ≥ 3.1.1. Theorem. A very general elliptic curve E over C(t) of height d ≥ 3 has the following property: For every finite rational extension L ∼ = C(u) of C(t), the Mordell-Weil group E(L) = 0.Here "very general" means that in the relevant moduli space, the statement holds on the complement of a countable union of proper closed subsets. See Subsection 3.4 below for more details.The theorem shows in particular that there is no hope of producing elliptic curves of large rank over C(u) by starting with a general curve over C(t) and iteratively making rational field extensions. What can be done with special elliptic curves remains a very interesting open question about which we make a few speculations in the last section. Now we connect the theorem with the title of the paper. Attached to E/C(t) is a unique elliptic surface π : E → P 1 with the properties that E is smooth over C, and π is proper and relatively minimal with generic fiber isomorphic to E/C(t). Conversely, given a relatively minimal elliptic surface π : E → P 1 , its generic fiber is an elliptic curve over C(t). The height of E is then equal

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

Cited by 26 publications

References 27 publications

The Mathematics of Networks of Linear Systems

The Mathematics of Networks of Linear Systems

Diophantine equations in separated variables

Rational curves on elliptic surfaces

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