Periodica Mathematica Hungarica

, Volume 78, Issue 1, pp 98–109

# Bounding univariate and multivariate reducible polynomials with restricted height

• Artūras Dubickas
Article

## Abstract

Let $$d,H \geqslant 2$$, $$m, u \geqslant 0$$ be some integers satisfying $$m+u \leqslant d$$. Consider a set of univariate integer polynomials of degree d whose m coefficients for the highest powers of x and u coefficients for the lowest powers of x are fixed, whereas the remaining $$g=d-m-u+1$$ coefficients are all bounded by H in absolute value. We show that among those $$(2H+1)^g$$ polynomials at most $$c d(2H+1)^{g-1}(\log (2H))^{\delta }$$ are reducible over $$\mathbb Q$$, where the constant $$c>0$$ depends only on two extreme coefficients (if they are fixed) and does not depend on d and H. Here, $$\delta =2$$ if $$m=u=0$$; $$\delta =1$$ if only one of mu is zero; $$\delta =0$$ if none of mu is zero. This estimate is better than the previous one in certain range of d and H. We also prove an estimate for the number of integer reducible polynomials in $$n \geqslant 2$$ variables of degree $$d \geqslant 1$$ in each variable and height at most $$H \geqslant 1$$. It is completely explicit in terms of ndH and implies that the probability for such a polynomial to be reducible tends to zero as $$\max (n,d,H) \rightarrow \infty$$. The condition $$n \geqslant 2$$ is essential in the proof: despite some recent progress the problem in general remains open for $$n=1$$.

## Keywords

Reducible polynomials Multivariate reducible polynomials Height

## Mathematics Subject Classification

11R09 12D05 12E05

## Notes

### Acknowledgements

This research was funded by a Grant (No. S-MIP-17-66/LSS-110000-1274) from the Research Council of Lithuania.

## References

1. 1.
S. Akiyama, A. Pethő, On the distribution of polynomials with bounded roots I. Polynomials with real coefficients. J. Math. Soc. Jpn. 66, 927–949 (2014)
2. 2.
S. Akiyama, A. Pethő, On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients. Unif. Distrib. Theory 9, 5–19 (2014)
3. 3.
L. Bary-Soroker, G. Kozma, Is a bivariate polynomial with $$\pm 1$$ coefficients irreducible? very likely!. Int. J. Number Theory 13, 933–936 (2017)
4. 4.
L. Bary-Soroker, G. Kozma, Irreducible polynomials of bounded height, preprint arXiv:1710.05165 (2017)
5. 5.
M. Bhargava, J.E. Cremona, T. Fisher, N.G. Jones, J.P. Keating, What is the probability that a random integral quadratic form in n variables has an integral zero? Int. Math. Res. Not. IMRN 12, 3828–3848 (2016)
6. 6.
F. Calegari, Z. Huang, Counting Perron numbers by absolute value. J. Lond. Math. Soc. 96, 181–200 (2017)
7. 7.
A. Castillo, R. Dietmann, On Hilbert’s irreducibility theorem. Acta Arith. 180, 1–14 (2017)
8. 8.
R. Chela, Reducible polynomials. J. Lond. Math. Soc. 38, 183–188 (1963)
9. 9.
S.-J. Chern, J.D. Vaaler, The distribution of values of Mahler’s measure. J. Reine Angew. Math. 540, 1–47 (2001)
10. 10.
K. Dörge, Abschätzung der Anzahl der reduziblen Polynome. Math. Ann. 160, 59–63 (1965)
11. 11.
A. Dubickas, Polynomials irreducible by Eisenstein’s criterion. Appl. Algebra Eng. Commun. Comput. 14, 127–132 (2003)
12. 12.
A. Dubickas, On the number of reducible polynomials of bounded naive height. Manuscr. Math. 144, 439–456 (2014)
13. 13.
A. Dubickas, Counting integer reducible polynomials with bounded measure. Appl. Anal. Discrete Math. 10, 308–324 (2016)
14. 14.
A. Dubickas, M. Sha, Counting and testing dominant polynomials. Exp. Math. 24, 312–325 (2015)
15. 15.
R. Grizzard, J. Gunther, Slicing the stars: counting algebraic integers, and units by degree and height. Algebra Number Theory 11, 1385–1436 (2017)
16. 16.
R. Heyman, I.E. Shparlinski, On the number of Eisenstein polynomials of bounded height. Appl. Algebra Eng. Commun. Comput. 24, 149–156 (2013)
17. 17.
R. Heyman, I.E. Shparlinski, On shifted Eisenstein polynomials. Period. Math. Hung. 69, 170–181 (2014)
18. 18.
S.V. Konyagin, On the number of irreducible polynomials with $$0,1$$ coefficients. Acta Arith. 88, 333–350 (1999)
19. 19.
F. Koyuncu, F. Özbudak, Probabilities for absolute irreducibility of multivariate polynomials by the polytope method. Turkish J. Math. 35, 367–377 (2011)
20. 20.
G. Kuba, On the distribution of reducible polynomials. Math. Slovaca 59, 349–356 (2009)
21. 21.
G. Micheli, R. Schnyder, The density of shifted and affine Eisenstein polynomials. Proc. Am. Math. Soc. 144, 4651–4661 (2016)
22. 22.
A.M. Odlyzko, B. Poonen, Zeros of polynomials with $$0,1$$ coefficients. Enseign. Math. 39, 317–348 (1993)
23. 23.
I. Rivin, Galois groups of generic polynomials, preprint arXiv:1511.06446 (2015)
24. 24.
G. Pólya, G. Szegö, Problems and Theorems in Analysis, vol. II (Springer, Berlin, 1976)
25. 25.
B.L. van der Waerden, Die Seltenhen der Gleichungen mit Affekt. Math. Ann. 109, 13–16 (1934)