Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-23T20:19:33.763Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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

REFERENCES

Cojocaru, A. C. and Murty, M. R.. An Introduction to Sieve Methods and their Applications (Cambridge University Press, 2005).10.1017/CBO9780511615993CrossRefGoogle Scholar
Estermann, T.. Einige Sätze über quadratfreie Zahlen. Math. Ann. 105 (1931), 653–662.CrossRefGoogle Scholar
Filaseta, M. and Trifonov, O.. On gaps between squarefree numbers II. J. London Math. Soc. s2-45 (1992), 215–221.CrossRefGoogle Scholar
Granville, A.. ABC allows us to count squarefrees. Int. Math. Res. Not. 19 (1998), 991–1009.CrossRefGoogle Scholar
Hooley, C.. On the square-free values of cubic polynomials. J. Reine Angew. Math. 229 (1968), 147–154.Google Scholar
Hooley, C.. Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts in Mathematics vol. 70 (Cambridge University Press, 1976).Google Scholar
Lando, G.. Square-free values of polynomials evaluated at primes over a function field. Q. J. Math. 66 (2015), 905–924.CrossRefGoogle Scholar
Poonen, B.. Squarefree values of multivariable polynomials. Duke Math. J. 118 (2003), 353–373.CrossRefGoogle Scholar
Ramsay, K.. Square-free values of polynomials in one variable over function fields. Int. Math. Res. Not. 4 (1992), 97–102.10.1155/S1073792892000114CrossRefGoogle Scholar
Ricci, G.. Ricerche aritmetiche sui polinomi. Rend. Circ. Mat. Palermo 57 (1933), 433–475.CrossRefGoogle Scholar
Tolev, D. I.. On the distribution of r-tuples of squarefree numbers in short intervals. Int. J. Number Theory 2 (2006), 225–234.CrossRefGoogle Scholar