On equal values of power sums of arithmetic progressions

Bazsó, András; Kreso, Dijana; Luca, Florian

doi:10.3336/gm.47.2.02

Cited by 11 publications

(15 citation statements)

References 13 publications

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“…Let γ 0 and γ 1 be two distinct roots of g . We now show that g(γ 0 ) = g(γ 1 ). Assume to the contrary that g(γ 0 ) = g(γ 1 ) = γ .…”

Section: Lemma 313 Let K Be a Field With Char(k ) = 0 And Let F ∈ K mentioning

confidence: 99%

“…If (1.3) holds, then k 0 t = l 0 t = 2, and thus t = 2 (since t > 1 by assumption), k 0 = l 0 = 1 and deg h = k = 2, a contradiction. If (1.4) holds, then t 0 = t 1 = 1 and either 2t 0 + 1 = 1 or 2t 1 …”

Section: Has Infinitely Many Solutions X Y With a Bounded O S -Denommentioning

confidence: 99%

“…In various papers, of which we mention [1,3,17], equations of type μ 1 (B k (μ 2 (x))) = λ 1 (B n (λ 2 (x))), and μ 1 (E k (μ 2 (x))) = λ 1 (E n (λ 2 (x))), where μ i , λ i ∈ Q[x] are linear and k, n ≥ 3, have been studied, corresponding to equations with the above introduced power sums. We do not have a proof at hand, but if Bernoulli and Euler polynomials are such that they have equal critical values at at most two distinct critical points, then Theorem 4.2 would yield a unifying proof of the results in these papers.…”

Section: Theorem 56mentioning

confidence: 99%

“…A defining equation of an elliptic curve is a prominent example of such equations. By Siegel's classical theorem, it follows that an irreducible algebraic curve defined over a B Dijana Kreso kreso@math.tugraz.at Robert F. Tichy tichy@tugraz.at 1 Graz University of Technology, Steyrergasse 30/II, 8010 Graz, Austria number field has only finitely many S-integral points, unless it has genus zero and no more than two points at infinity. Ever since Siegel's theorem, one of the driving questions was to classify polynomials f, g for which the equation f (x) = g(y) has infinitely many solutions in S-integers x, y.…”

Section: Introductionmentioning

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.

show abstract

“…Let γ 0 and γ 1 be two distinct roots of g . We now show that g(γ 0 ) = g(γ 1 ). Assume to the contrary that g(γ 0 ) = g(γ 1 ) = γ .…”

Section: Lemma 313 Let K Be a Field With Char(k ) = 0 And Let F ∈ K mentioning

confidence: 99%

Section: Has Infinitely Many Solutions X Y With a Bounded O S -Denommentioning

confidence: 99%