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On square-free values of large polynomials over the rational function field

Part of: Algebraic number theory: global fields Multiplicative number theory Curves

Published online by Cambridge University Press:  12 December 2019

DAN CARMON  and
ALEXEI ENTIN
DAN CARMON
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. e-mails: dancarmo@post.tau.ac.il; aentin@tauex.tau.ac.il
ALEXEI ENTIN
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. e-mails: dancarmo@post.tau.ac.il; aentin@tauex.tau.ac.il
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Abstract

We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11N36: Applications of sieve methods 11R58: Arithmetic theory of algebraic function fields 14H05: Algebraic functions; function fields
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 247 - 263
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° 320755.

References

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