How does the discriminant of integer polynomials depend on the distribution of roots?

N. V. Budarina ; V. I. Bernik ; H. O'Donnell

Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 153-162.

Résumé

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be some natural parameter, and $\mathcal{P}_n(Q)$ denote the set of integer polynomials of degree $n$ and height of at most $Q$. Given a polynomial $P(x)=a_nx^n+\cdots+a_0\in\mathbb{Z}[x]$ of degree $n$, the discriminant of $P(x)$ is defined by $$ D(P)=a_n^{2n-2}\prod_{1\le i\le n}(\alpha_i-\alpha_j)^2, $$ where $\alpha_1, \ldots, \alpha_n\in\mathbb{C}$ are the roots of $P(x)$. In this paper we investigate the following problem on the number of polynomials with small discriminants: for a given $0\le v\le 2$ and sufficiently large $Q$, estimate the value of $\#\mathcal{P}_n(Q,v) $, where $\mathcal{P}_n(Q,v)$ denote the class of polynomials $P\in\mathcal{P}_n(Q)$ such that $$ 0|D(P)|\le Q^{2n-2-2v}. $$ The first results for the estimate of the number of polynomials with given discriminants were received by H. Davenport in 1961, which were crucial to the solving of the problem of Mahler. In this paper for the first time we obtain the exact upper and lower bounds for $\#\mathcal{P} _3(Q,v)$ with the additional condition on the distribution of the roots of the polynomials. It is interesting that the value of $\#\mathcal{P}_n(Q,v)$ has the largest value when all the roots of polynomials are close to each other. If there are only $k$, $2\le k$, close roots to each other then the value of $\#\mathcal{P}_n(Q,v)$ will be less. Bibliography: 15 titles.

Détail

Keywords: integer polynomials, approximation by algebraic numbers, discriminants of polynomials.

Bibliographie

[1] Van Der Waerden B. L., Algebra, Springer-Verlag, Berlin–Heidelberg, 1971

[2] Davenport H., “A note on binary cubic forms”, Mathematika, 8 (1961), 58–62 | DOI

[3] Sprindzhuk V. G., Mahler's problem in metric theory of numbers, Minsk, 1967 (Russian)

[4] Beresnevich V., “On approximation of real numbers by real algebraic numbers”, Acta Arithmetica, 90:2 (1999), 97–112

[5] Bernik V. I., “The exact order of approximating zero by values of integral polynomials”, Acta Arithmetica, 53:1 (1989), 17–28

[6] Bernik V. I., “Application of the Hausdorff dimension in the theory of Diophantine approximations”, Acta Arithmetica, 42:3 (1983), 219–253

[7] Budarina N., Dickinson D., Bernik V., “A divergent Khintchine theorem in the real, complex and $p$-adic fields”, Lith. Math. J., 48:2 (2008), 158–173 | DOI

[8] Budarina N., Dickinson D., Bernik V., “Simultaneous Diophantine approximation in the real, complex and $p$-adic fields”, Math. Proc. Cambridge Philos. Soc., 149:2 (2010), 193–216 | DOI

[9] Goetze F., Kaliada D., Korolev M., “On the number of quadratic polynomials with bounded discriminants”, Mat. Zametki (to appear)

[10] Goetze F., Kaliada D., Kukso O., “The asymptotic number of integral cubic polynomials with bounded heights and discriminants”, Lith. Math. J., 54:2 (2014), 150–165 | DOI

[11] Bernik V., Goetze F., Kukso O., “Lower bounds for the number of integral polynomials with given order of discriminants”, Acta Arithmetica, 133:4 (2008), 375–390 | DOI

[12] Beresnevich V. V., Bernik V. I., Goetze F., “Simultaneous approximations of zero by an integral polynomial, its derivative, and small values of discriminants”, Dokl. Nats. Nauk Belarusi, 54:2 (2010), 26–28; 125

[13] Beresnevich V., “Rational points near manifolds and metric Diophantine approximation”, Ann. of Math., 175:1 (2012), 187–235 | DOI

[14] Bernik V., Budarina N., “On arithmetic properties of integral polynomials with small values on the interval”, Siauliai Math. Semin., 8:16 (2013), 27–36

[15] Koleda D. V., “An upper bound for the number of integral polynomials of third degree with a given bound for discriminants”, Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, 2010, no. 3, 10–16; 124