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On some issues concerning polynomial cycles

Tadeusz Pezda (2013)

Communications in Mathematics

We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain R of positive characteristic (for N 1 ) or for any Dedekind domain R of positive characteristic (but only for N 2 ), we give a closed formula for a set 𝒞 Y C L ( R , N ) of all possible cycle-lengths for polynomial mappings in R N . Then we give a new property of sets 𝒞 Y C L ( R , 1 ) , which refutes a kind of conjecture posed by W. Narkiewicz.

On the Győry-Sárközy-Stewart conjecture in function fields

Igor E. Shparlinski (2018)

Czechoslovak Mathematical Journal

We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product ( a b + 1 ) ( a c + 1 ) ( b c + 1 ) for distinct positive integers a , b and c . In particular, we show that, under some natural conditions on rational functions F , G , H ( X ) , the number of distinct zeros and poles of the shifted products F H + 1 and G H + 1 grows linearly with deg H if deg H max { deg F , deg G } . We also obtain a version of this result for rational functions over a finite field.

On the irreducibility of 0,1-polynomials of the form f(x)xⁿ + g(x)

Michael Filaseta, Manton Matthews, Jr. (2004)

Colloquium Mathematicae

If f(x) and g(x) are relatively prime polynomials in ℤ[x] satisfying certain conditions arising from a theorem of Capelli and if n is an integer > N for some sufficiently large N, then the non-reciprocal part of f(x)xⁿ + g(x) is either identically ±1 or is irreducible over the rationals. This result follows from work of Schinzel in 1965. We show here that under the conditions that f(x) and g(x) are relatively prime 0,1-polynomials (so each coefficient is either 0 or 1) and f(0) = g(0) = 1, one...

On the irreducible factors of a polynomial over a valued field

Anuj Jakhar (2024)

Czechoslovak Mathematical Journal

We explicitly provide numbers d , e such that each irreducible factor of a polynomial f ( x ) with integer coefficients has a degree greater than or equal to d and f ( x ) can have at most e irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field.

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