Polynomial cycles in finite extension fields
Polynomial mappings defined by forms with a common factor
Polynomial orbits in finite commutative rings
Let be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.
Polynomials dividing infinitely many quadrinomials or quintinomials
Polynomials generated by the Fibonacci sequence.
Polynomials whose powers are sparse
Polynomials whose values are powers.
Polynomials with integral coefficients, equivalent to a given polynomial.
Powers of a matrix and combinatorial identities.
Poznámka o celočíselných polynomech, jejichž hodnoty jsou dělitelné číslem
Článek si dává za cíl ukázat, že z kanonických polynomů Dn(x) lze pomocí určitých lineárních kombinací vytvořit všechny polynomy, které jsou dělitelné n!. Autor formuluje větu o dělitelnosti těchto polynomů n!. Z této věty pak vyplývá celá řada tvrzení, z kterých uvádí pouze prvních šest. V každém tvrzení nalezne polynom a postupně tvrdí, že první je dělitelný 2, další 6, další 24, další číslem 120, další 720 a poslední 5040 pro celočíselné koeficienty. Vzhledem k těmto tvrzením formuluje obecné...
PPCM de suites de polynômes
Prime factors of values of polynomials
We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.
Prime rational functions
Let f(x) be a complex rational function. We study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.
Prime values of irreducible polynomials
Prime values of reducible polynomials, I
Prime values of reducible polynomials, II
Primteiler von Polynomen.
Reducibility and irreducibility of Stern -polynomials
The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials defined by , , , and ; these polynomials are Newman polynomials, i.e., they have only 0 and 1 as coefficients. In this paper we prove numerous reducibility and irreducibility properties of these polynomials, and we show that cyclotomic polynomials play an important role as factors. We also prove several related results, such as the fact that can only have simple zeros, and we state a...
Reducibility of lacunary polynomials II
Reducibility of lacunary polynomials, VI