On the existence of solutions of a Fermat-type difference equation

Li, Nan

doi:10.5186/aasfm.2015.4051

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…author [10], who obtained similar bounds for a difference counterpart of (1.2) under certain conditions on the value distribution of solutions.…”

Section: Introductionmentioning

confidence: 77%

Mason's theorem with a difference radical

Ishizaki¹,

Korhonen²,

Li³

et al. 2018

Preprint

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Differential calculus is not a unique way to observe polynomial equations such as a+b = c. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Two results on the non-existence of polynomial solutions to difference Fermat type functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider difference Fermat type equations with transcendental entire solutions.

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“…author [10], who obtained similar bounds for a difference counterpart of (1.2) under certain conditions on the value distribution of solutions.…”

Section: Introductionmentioning

confidence: 77%

Mason's theorem with a difference radical

Ishizaki¹,

Korhonen²,

Li³

et al. 2018

Preprint

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“…Hayman [18] calls the problem of finding the smallest m = G 0 (n) for which a solution of (1.2) exists as the Super-Fermat problem. A difference analogue of (1.2) was studied by the third author [26], who obtained similar bounds for a difference counterpart of (1.2) under certain conditions on the value distribution of solutions. The purpose of this paper is to introduce a difference counterpart of the radical, and to use it to prove a difference analogue of the Stothers-Mason theorem, as well as a truncated version of the difference second main theorem for holomorphic curves.…”

Section: Introductionmentioning

confidence: 87%

A Stothers–Mason theorem with a difference radical

Ishizaki

Korhonen

et al. 2020

Math. Z.

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Differential calculus is not a unique way to observe polynomial equations such as $$a+b=c$$ a + b = c . We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Results, for example, on the non-existence of polynomial solutions to difference Fermat and difference Super-Fermat functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider these functional equations with non-polynomial entire solutions. Equations with polynomial or non-polynomial solutions are observed to see the sharpness of results obtained.

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“…We study a difference analogue of radical of a complex polynomial, and consider a difference analogue of the Stothers-Mason theorem. As applications, we study the difference version of the Fermat type functional equations, see e.g., [4], [9].…”

Section: Introductionmentioning

confidence: 99%