On the number of polynomials of bounded measure
On the product of heights of algebraic numbers summing to real numbers
On the product of the conjugates outside the unit circle of an algebraic number
On the reducibility type of trinomials
On the relation between two conjectures on polynomials
Patterns and periodicity in a family of resultants
Given a monic degree polynomial and a non-negative integer , we may form a new monic degree polynomial by raising each root of to the th power. We generalize a lemma of Dobrowolski to show that if and is prime then divides the resultant of and . We then consider the function . We show that for fixed and that this function is periodic in both and , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.
Periodic points, multiplicities, and dynamical units.
Points périodiques des automorphismes affines.
Pólya fields and Pólya numbers
A number field , with ring of integers , is said to be a Pólya field if the -algebra formed by the integer-valued polynomials on admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of in a Pólya field. We give a positive answer to this embedding problem by showing that...
Pólya fields, Pólya groups and Pólya extensions: a question of capitulation
A number field , with ring of integers , is said to be a Pólya field when the -algebra formed by the integer-valued polynomials on admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when is not a Pólya field, we are interested in the embedding of in a Pólya field. We study here two notions which can be considered as measures...
Polynome mit nicht durch Restklassen beschreibbaren Primteilermengen
Polynômes à valeurs entières
Polynomial cycles in certain rings of rationals
It is shown that the methods established in [HKN3] can be effectively used to study polynomial cycles in certain rings. We shall consider the rings and shall describe polynomial cycles in the case when is either odd or twice a prime.
Polynomial mappings defined by forms with a common factor
Polynomial relations amongst algebraic units of low measure
For an algebraic number field and a subset , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in vanishing at the point .
Polynomials in many variables
Polynomials of the form and pseudoprimes with respect to linear recurring sequences
Polynomials that divide many trinomials
Polynomials whose values are powers.
Polynomials with height 1 and prescribed vanishing at 1.