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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.

##### MSC:
 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)
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