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Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

Given a monic degree N polynomial f ( x ) ∈ ℤ [ x ] and a non-negative integer ℓ , we may form a new monic degree N polynomial f ℓ ( x ) ∈ ℤ [ x ] by raising each root of f to the ℓ th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) ↦ Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

Pólya fields and Pólya numbers

Amandine Leriche (2010)

Actes des rencontres du CIRM

A number field K , with ring of integers 𝒪 K , is said to be a Pólya field if the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K 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 K is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of K 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

Amandine Leriche (2011)

Journal de Théorie des Nombres de Bordeaux

A number field K , with ring of integers 𝒪 K , is said to be a Pólya field when the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K 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 K is not a Pólya field, we are interested in the embedding of K in a Pólya field. We study here two notions which can be considered as measures...

Polynomial cycles in certain rings of rationals

Władysław Narkiewicz (2002)

Journal de théorie des nombres de Bordeaux

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 𝐙 [ 1 N ] and shall describe polynomial cycles in the case when N is either odd or twice a prime.

Polynomial relations amongst algebraic units of low measure

John Garza (2014)

Acta Arithmetica

For an algebraic number field and a subset α 1 , . . . , α r ⊆ , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in ℚ [ x 1 , . . . , x r ] vanishing at the point ( α 1 , . . . , α r ) .

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