On the abc conjecture in algebraic number fields

Győry, Kálmán

doi:10.4064/aa133-3-6

Cited by 20 publications

(37 citation statements)

References 23 publications

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“…Our proof of the decidability of ABCD-Z problem for 2 × 2 upper-triangular rational matrices relies on the following result which is proved using Baker's theorem on linear forms in logarithms (see Corollary 4 in [16] and also [18]). Theorem 5.…”

Section: On the Mortality Problemmentioning

confidence: 99%

On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

Bell

Potapov

Semukhin

2021

Information and Computation

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We consider the following variant of the Mortality Problem: given k × k matrices A1, A2, . . . , At, does there exist nonnegative integers m1, m2, . . . , mt such that the productis equal to the zero matrix? It is known that this problem is decidable when t ≤ 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices.In this paper, we prove the first decidability results for t > 2. We show as one of our central results that for t = 3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t = 3 and k ≤ 3 for matrices over algebraic numbers and for t = 3 and k = 4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m1, m2, m3) for which the equation A m 1 1 A m 2 2 A m 3 3 equals the zero matrix is equal to a finite union of direct products of semilinear sets.For t = 4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2×2 rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations. ACM Subject ClassificationTheory of computation → Computability; Mathematics of computing → Computations on matrices

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Section: On the Mortality Problemmentioning

confidence: 99%

On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

Bell

Potapov

Semukhin

2021

Information and Computation

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“…The abc conjecture has several applications, the reader may refer to [9], [3], [6], [7] for details.…”

Section: The Abc-conjecturementioning

confidence: 99%

“…To state the analogue of abc-conjecture for number fields, we need some preparations, which we do below. The interested reader may refer to [9], [3] for more details.…”

Section: The Abc-conjecturementioning

confidence: 99%

Non-Wieferich primes in number fields and $abc$-conjecture

Kotyada¹,

Muthukrishnan²

2018

Czech.Math.J.

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“…In [Győ08], one may take β 1 = 1 + ǫ. Thus the bound obtained in Theorem 4.2 is less good as the known results.…”

Section: Conjecture 41 (Masser-oesterlé) (Abc)mentioning

confidence: 99%

“…Using such a bound given by a quantitative Siegel's theorem due to Bilu [Bil97], the second author obtained in her thesis (see [Sur07]) the first result towards the abc-conjecture over an arbitrary number field. Afterwards, K. Gyory and K. Yu ([GY06], [Győ08]) gave completely explicit abc-type results using bounds for the height of solutions of S-unit equations. When the number field is Q better inequalities were known, see [ST86] for the first result obtained and [SY91], [SY01] for later improvements.…”

Section: Introductionmentioning

confidence: 99%

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Bosser

Surroca

2014

Bull Braz Math Soc, New Series

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Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.

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On the abc conjecture in algebraic number fields

Cited by 20 publications

References 23 publications

On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

Non-Wieferich primes in number fields and $abc$-conjecture

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

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