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Shifted Power of a Polynomial with Integral Roots

  • Artūras Dubickas
Published/Copyright: August 4, 2023
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ABSTRACT

In this note, for any integers m, n ≥ 2, we find a condition on a positive integer c under which there exists a monic polynomial f[x] of degree n for which f(x) m c has mn integral roots counting with multiplicities. This is the case if and only if m = 2 and c is a constant that comes from a solution of the Prouhet-Tarry-Escott problem of size n. For example, the smallest positive integer c for which there exists a monic degree 7 polynomial f[x] such that f(x)2c has 14 integral roots is c = 6620176679276160000.

2020 Mathematics Subject Classification: Primary 12D05; Secondary 11D72; 11A05

(Communicated by István Gaál)


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Received: 2022-02-22
Accepted: 2022-09-11
Published Online: 2023-08-04

© 2023 Mathematical Institute Slovak Academy of Sciences

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