Wyniki wyszukiwania

1

Cycles of polynomials in algebraically closed fields of positive characteristic

100%

Pezda T.

Colloquium Mathematicum

|

1994

|

tom 67

|

nr 2

187-195

2

Polynomial cycles in certain local domains

100%

Pezda T.

Acta Arithmetica

|

1994

|

tom 66

|

nr 1

11-22

EN

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x₀,x₁,...,x_{k-1}$ of distinct elements of R is called a cycle of f if $f(x_i) = x_{i+1}$ for i=0,1,...,k-2 and $f(x_{k-1}) = x₀$. The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number $7^{7·2^N}$, depending only on the degree N of K. In this note we consider the case when R is a discrete valuation domain of zero characteristic with finite residue field. We shall obtain an upper bound for the possible lengths of cycles in R and in the particular case R=ℤₚ (the ring of p-adic integers) we describe all possible cycle lengths. As a corollary we get an upper bound for cycle lengths in the ring of integers in an algebraic number field, which improves the bound given in [1]. The author is grateful to the referee for his suggestions, which essentially simplified the proof in Subsection 6 and improved the bound for C(p) in Theorem 1 in the case p = 2,3.

3

Cycles of polynomials in algebraically closed fields of positive characteristic (II)

100%

Pezda T.

Colloquium Mathematicum

|

1996

|

tom 71

|

nr 1

23-30

4

Cycles of polynomial mappings in two variables over rings of integers in quadratic fields

100%

Pezda T.

Open Mathematics

|

2004

|

tom 2

|

nr 2

294-331

EN

We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals.

5

Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals

100%

Pezda T.

Acta Arithmetica

|

2003

|

tom 108

|

nr 2

127-146

6

On prime values of reducible quadratic polynomials

64%

Narkiewicz W., Pezda T.

Colloquium Mathematicum

|

2002

|

tom 93

|

nr 1

151-154

EN

It is shown that Dickson's Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_r$ such that the polynomial $f(X)/N_r$ represents at least r distinct primes.

Ograniczanie wyników

3 Colloquium Mathematicum

2 Acta Arithmetica

1 Open Mathematics

6 Pezda T.

1 Narkiewicz W.

1 2004

1 2003

1 2002

1 1996

2 1994

Cycles of polynomials in algebraically closed fields of positive characteristic

Polynomial cycles in certain local domains

Cycles of polynomials in algebraically closed fields of positive characteristic (II)

Cycles of polynomial mappings in two variables over rings of integers in quadratic fields

Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals

On prime values of reducible quadratic polynomials