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On the distribution of roots of polynomials in sectors. I. (English) Zbl 0920.12001
Lith. Math. J. 38, No. 1, 27-45 (1998); reprinted from Liet. Mat. Rink. 38, No. 1, 34-58 (1998).
Let \(P(X)=a_nX^n+\cdots+a_0\) (\(a_0,a_n\neq 0\)) be a complex polynomial. For \(0\leq\psi\leq\varphi\leq 2\pi\) denote by \(N_P(\psi,\varphi)\) the number of roots \(z\) of \(P\) satisfying \(\psi\leq\arg z\leq\varphi\), and put \[ E_P=E_P(\psi,\varphi)=\biggl| N_P(\psi,\varphi)-{(\psi-\varphi)n\over 2\pi} \biggr|. \] It was shown by P. Erdős and P. Turán [Ann. Math. (2) 51, 105-119 (1950; Zbl 0036.01501)] that if \(L(P)\) denotes the sum of absolute values of coefficients of \(P\), then \[ E_P<16\left(n\log\left({L(P)\over a_na_0}\right)\right)^{1/2}. \] T. Ganelius [Ark. Mat. 3, 1-50 (1954; Zbl 0055.06905)] replaced the coefficient \(16\) by \(2.61\dots\).
Later an upper bound for \(E_P\) involving Mahler’s measure of \(P\) was proved by M. Mignotte [Acta Arith. 54, 81-86 (1989; Zbl 0641.12003)].
The author considers the case \(n\geq 10\) and obtains (Theorem 1) a bound of the form \(E_P<C\sqrt n\log n\), where \(C\) depends on \(P\) in the following way: \(C=\sqrt{A+2B+2}+2\sqrt B + 1\), where \[ A={(2n-2)\log| a_n| -\log| \Delta(P)| \over n\log n} \] and \[ {n\over 10\log n}\geq B\geq{n\log d(P)\over\log n}, \] \(D(P)\) being the discriminant of \(P\) and \(d(P)=\max\{| z| ,1/| z| :P(z)=0\}\) denoting the symmetric deviation from the unit circle of roots of \(P\). In the case when \(P\) has integral coefficients and is not a reciprocal polynomial this bound can be improved (Theorem 2). Moreover bounds for \(E_P\) are obtained in terms of the discriminant and the logarithm of Mahler’s measure.
For Part II, see the review Zbl 0920.12002 below.

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
Full Text: DOI
[1] F. Amoroso, Algebraic numbers close to 1 and variants of Mahler’s measure,J. Number Theory,60, 80–96 (1996). · Zbl 0866.11060 · doi:10.1006/jnth.1996.0114
[2] Yu. Bilu (Belotserkovski), Uniform distribution of algebraic numbers near the unit circle,Vesti Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk,124 (1), 49–52 (1988).
[3] Yu. Bilu, Limit distribution of small points on algebraic tori,Duke Math. J. (to appear).
[4] D. W. Boyd, Reciprocal polynomials having small measure,Math. Comp.,35, 1361–1377 (1980). · Zbl 0447.12002 · doi:10.1090/S0025-5718-1980-0583514-9
[5] D. W. Boyd, Reciprocal polynomials having small measure. II,Math. Comp.,53, 355–357 (1989). · Zbl 0684.12003 · doi:10.1090/S0025-5718-1989-0968149-6
[6] A. Dubickas, Algebraic conjugates outside the unit circle, in:New Trends in Probability and Statistics. Vol. 4. Analytic and Probabilistic Methods in Number Theory.Proceedings of the Second Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 23–27 September, 1996, A. Laurinčikas, E. Manstavičius, and V. Stak\.enas (Eds.), VSP, Utrecht/ TEV, Vilnius (1997), pp. 11–21.
[7] P. Erdös and P. Turán, On the distribution of roots of polynomials,Ann. Math.,51, 105–119 (1950). · Zbl 0036.01501 · doi:10.2307/1969500
[8] T. Ganelius, Sequences of analytic functions and their zeros,Ark. Mat.,3, 1–50 (1954). · Zbl 0055.06905 · doi:10.1007/BF02589280
[9] M. Langevin, Méthode de Fekete-Szegö et problème de Lehmer,C. R. Acad. Sci. Paris,301, 463–466 (1985). · Zbl 0585.12013
[10] C. W. Lloyd-Smith, Algebraic numbers near the unit circle,Acta Arith.,43, 43–57 (1985). · Zbl 0533.12002
[11] M. Mignotte, Sur un théorème de M. Langevin,Acta Arith.,54, 81–86 (1989). · Zbl 0641.12003
[12] M. Mignotte, Remarque sur une question relative à des fonctions conjuguées,C. R. Acad. Sci. Paris,315, 907–911 (1992). · Zbl 0773.31002
[13] M. Mossinghoff,Algorithms for the determination of polynomials with small Mahler measure, Ph.D. Thesis, University of Texas at Austin (1995).
[14] C. Notari, Sur le produit des conjugués à l’extérieur du circle unité d’un nombre algébrique,C. R. Acad. Sci. Paris,286, 313–315 (1978). · Zbl 0387.30003
[15] G. Rhin and C. J. Smyth, On the Mahler measure of polynomials having all zeros in a sector,Math. Comp.,209, 295–304 (1995). · Zbl 0820.11064 · doi:10.1090/S0025-5718-1995-1257579-6
[16] C. J. Smyth, On the product of conjugates outside the unit circle of an algebraic integer.Bull. London Math. Soc.,3, 169–175 (1971). · Zbl 0235.12003 · doi:10.1112/blms/3.2.169
[17] P. Voutier, An effective lower bound for the height of algebraic numbers,Acta Arith.,74, 81–95 (1996). · Zbl 0838.11065
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