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

Published online by Cambridge University Press:  12 December 2019

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

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.

Type
Research Article
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

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), 653662.CrossRefGoogle Scholar
Filaseta, M. and Trifonov, O.. On gaps between squarefree numbers II. J. London Math. Soc. s2-45 (1992), 215221.CrossRefGoogle Scholar
Granville, A.. ABC allows us to count squarefrees. Int. Math. Res. Not. 19 (1998), 9911009.CrossRefGoogle Scholar
Hooley, C.. On the square-free values of cubic polynomials. J. Reine Angew. Math. 229 (1968), 147154.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), 905924.CrossRefGoogle Scholar
Poonen, B.. Squarefree values of multivariable polynomials. Duke Math. J. 118 (2003), 353373.CrossRefGoogle Scholar
Ramsay, K.. Square-free values of polynomials in one variable over function fields. Int. Math. Res. Not. 4 (1992), 97102.10.1155/S1073792892000114CrossRefGoogle Scholar
Ricci, G.. Ricerche aritmetiche sui polinomi. Rend. Circ. Mat. Palermo 57 (1933), 433475.CrossRefGoogle Scholar
Tolev, D. I.. On the distribution of r-tuples of squarefree numbers in short intervals. Int. J. Number Theory 2 (2006), 225234.CrossRefGoogle Scholar