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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.
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...
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...
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.
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 .
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