Algebraic numbers as product of powers of transcendental numbers

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F19%3A50015674" target="_blank" >RIV/62690094:18470/19:50015674 - isvavai.cz</a>
Result on the web
<a href="https://www.mdpi.com/2073-8994/11/7/887" target="_blank" >https://www.mdpi.com/2073-8994/11/7/887</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.3390/sym11070887" target="_blank" >10.3390/sym11070887</a>

Result language
angličtina
Original language name
Algebraic numbers as product of powers of transcendental numbers
Original language description
The elementary symmetric functions played a crucial role in the study of zeros of non-zero polynomials in $C[x]$, and the problem of finding zeros in $Q[x]$ leads to the definition of algebraic and transcendental numbers. Recently, [Marques, D. Algebraic numbers of the form $P(T)^{Q(T)}$, with $T$ transcendental, textit{Elem. Math.} {bf 2010}, {em 65}, 78--80.] studied the set of algebraic numbers in the form $P(T)^{Q(T)}$. In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form $P_1(T)^{Q_1(T)}cdots P_n(T)^{Q_n(T)}$ for some transcendental number $T$, where $P_1,ldots,P_n,Q_1,ldots,Q_n$ are prescribed, non-constant polynomials in $Q[x]$ (under weak conditions). More generally, our result generalizes results on the arithmetic nature of $z^w$ when $z$ and $w$ are transcendental.
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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