Estimating the greatest common divisor of the value of two polynomials

Let \(p\) be a fixed prime, and let \(v(a)\) stand for the exponent of \(p\) in the prime factorization of the integer \(a\). Let \(f\) and \(g\) be two monic polynomials with integer coefficients and nonzero resultant \(r\). Write \(S\) for the maximum of \(v(\gcd (f(n), g(n)))\) over all integers...

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Bibliographic Details
Published in:arXiv.org 2017-12
Main Authors: Frenkel, Péter E, Zábrádi, Gergely
Format: Article
Language:English
Subjects:
Integers
Polynomials
Upper bounds
Online Access:Get full text
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Summary:Let \(p\) be a fixed prime, and let \(v(a)\) stand for the exponent of \(p\) in the prime factorization of the integer \(a\). Let \(f\) and \(g\) be two monic polynomials with integer coefficients and nonzero resultant \(r\). Write \(S\) for the maximum of \(v(\gcd (f(n), g(n)))\) over all integers \(n\). It is known that \(S \le v(r)\). We give various lower and upper bounds for the least possible value of \(v(r)-S\) provided that a given power \(p^s\) divides both \(f(n)\) and \(g(n)\) for all \(n\). In particular, the least possible value is \(ps^2-s\) for \(s\le p\) and is asymptotically \((p-1)s^2\) for large \(s\).
ISSN:2331-8422
DOI:10.48550/arxiv.1712.01054