On the Prime Divisors of Polynomials

Gerst, Irving; Brillhart, John

doi:10.1080/00029890.1971.11992737

Cited by 20 publications

(14 citation statements)

References 6 publications

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“…The following lemma is well-known and can be deduced from a result of Schur [GB71,Sch12]. Since we are unaware of a reference for the exact form that we need, we give a proof.…”

Section: Thenmentioning

confidence: 99%

A transcendental dynamical degree

2020

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“…The following lemma is well-known and can be deduced from a result of Schur [GB71,Sch12]. Since we are unaware of a reference for the exact form that we need, we give a proof.…”

Section: Thenmentioning

confidence: 99%

A transcendental dynamical degree

2020

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“…The following result from [13] will be useful for obtaining further properties of spectra, by exploiting the relation between irreducible polynomials and algebraic finite extensions of the rational field Q. (These extensions can be defined by adjoining to Q a root of a polynomial irreducible over Q.…”

Section: The Logic Of Finite Residue Classes and Circuit Complexitymentioning

confidence: 99%

“…[13, Thm. 2]) Let Q be the rational field and f (x) and g(x) two non constant irreducible polynomials in Q[x]…”

mentioning

confidence: 99%

OUP accepted manuscript

Arratia

Ortíz

2017

Journal of Logic and Computation

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“…In particular, Ax proved the following: Given a polynomial f (x) ∈ Z[x] we will indistinctly denote Sp(f ) or Sp(∃x(f (x) = 0)) the spectrum of the sentence ∃x(f (x) = 0). A basic result of Schur states that every non constant polynomial has an infinite number of prime divisors; that is, [12,Thm. 1] for an elementary proof of this fact).…”

Section: ) Dlogtime-uniformmentioning

confidence: 99%

Methods of Class Field Theory to Separate Logics over Finite Residue Classes and Circuit Complexity

Arratia,

Ortiz

2015

Preprint

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Separations among the first order logic Ring(0, +, * ) of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIMEuniform circuit complexity classes ranging from AC 0 to T C 0 . Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no h-density. One application of this result is that * Supported by MICINN project TIN2011-27479-C04-03 (BASMATI), MINECO project TIN2014-57226-P (APCOM) and Gen. Cat. project SGR2014-890 (MACDA).1 in Ring(0, +, * , <) + M , the logic of finite residue class rings with built-in order and extended with the majority quantifier M , there are sentences whose spectrum have no exponential density.

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On the Prime Divisors of Polynomials

Cited by 20 publications

References 6 publications

A transcendental dynamical degree

A transcendental dynamical degree

OUP accepted manuscript

Methods of Class Field Theory to Separate Logics over Finite Residue Classes and Circuit Complexity

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