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Multiplicative dependence of shifted algebraic numbers

100%

Drungilas P., Dubickas A.

Colloquium Mathematicum

2003

tom 96

nr 1

75-81

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

On degrees of three algebraic numbers with zero sum or unit product

100%

Drungilas P., Dubickas A.

Colloquium Mathematicum

2016

tom 143

nr 2

159-167

Let α, β and γ be algebraic numbers of respective degrees a, b and c over ℚ such that α + β + γ = 0. We prove that there exist algebraic numbers α₁, β₁ and γ₁ of the same respective degrees a, b and c over ℚ such that α₁ β₁ γ₁ = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a,b,c) ∈ ℕ³ for which there exist finite field extensions K/k and L/k (of a fixed field k) of degrees a and b, respectively, such that the degree of the compositum KL over k equals c. Towards another earlier formulated conjecture, under certain natural assumptions (related to the inverse Galois problem), we show that the set of such triplets forms a multiplicative semigroup.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Drungilas P.

2 Dubickas A.

1 2016

1 2003

Wyniki wyszukiwania

Multiplicative dependence of shifted algebraic numbers

On degrees of three algebraic numbers with zero sum or unit product