Zero Distribution of Polynomials

Abstract

The classical theorems of Jentzsch and Szegö concern the limiting behavior of the zeros of the partial sums of a power series. More precisely, if

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa % aaleaacaWGUbaabeaakmaabmaabaGaamOEaaGaayjkaiaawMcaaiab % g2da9maaqahabaGaamyyamaaBaaaleaacaWGRbaabeaakiaadQhada % ahaaWcbeqaaiaadUgaaaaabaGaam4Aaiabg2da9iaaicdaaeaacaWG % UbaaniabggHiLdGccaGGSaGaamOBaiabg2da9iaaicdacaGGSaGaaG % ymaiaacYcacaaIYaGaaiilaiablAciljaacYcaaaa!4E6B!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ {s_n}\left( z \right) = \sum\limits_{k = 0}^n {{a_k}{z^k}} ,n = 0,1,2, \ldots ,$$

are the partial sums of a power series EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaamOEaaGaayjkaiaawMcaaiabg2da9maaqaeabaWaa0baaSqa % aiaadUgacqGH9aqpcaaIWaaabaGaeyOhIukaaOGaamyyamaaBaaale % aacaWGRbaabeaakiaadQhadaahaaWcbeqaaiaadUgaaaaabeqab0Ga % eyyeIuoaaaa!44F6!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f\left( z \right) = \sum {_{k = 0}^\infty {a_k}{z^k}} $$ having finite positive radius of convergence ρ, then Jentzsch [91] proved that each point of the circle of convergence C _ρ := {z: |z| = ρ} is a limit point of zeros of polynomials s _n (z), n = 1, 2,... . Szegö [170] substantially improved this result by showing that there is a subsequence EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WGUbWaaSbaaSqaaiaadUgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa % aiaadUgacqGH9aqpcaaIXaaabaGaeyOhIukaaaaa!3E8D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {{n_k}} \right\}_{k = 1}^\infty $$ for which the zeros of the partial sums EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa % aaleaacaWGUbaabeaakmaaBaaaleaadaWgaaadbaGaam4Aaaqabaaa % leqaaOWaaeWaaeaacaWG6baacaGLOaGaayzkaaaaaa!3BFB!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${s_n}_{_k}\left( z \right)$$ are uniformly distributed in angle; that is, if S(α, β) is the sector

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaabm % aabaGaeqySdeMaaiilaiabek7aIbGaayjkaiaawMcaaiabg2da9maa % cmaabaGaamOEaiabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHr % hAGq1DVbacfaGae8NaHmKaaiOoaiabeg7aHjabgYda8iGacggacaGG % YbGaai4zaiaadQhacqGH8aapcqaHYoGyaiaawUhacaGL9baacaGGSa % GaeqySdeMaeyipaWJaeqOSdiMaeyipaWJaaGOmaiabec8aWjabgUca % Riabeg7aHbaa!614C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left( {\alpha ,\beta } \right) = \left\{ {z \in \mathbb{C}:\alpha < \arg z < \beta } \right\},\alpha < \beta < 2\pi + \alpha $$

, and Z _n (A) denotes the number of zeros of s _n in the set A, then

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGa % aeqabaWaaeaaeaaakeaacaaMe8UaaGjbVpaaxababaaeaaaaaaaaa8 % qacyGGSbaBcqGGPbqAcqGGTbqBaSWdaeaapeGaem4AaSMaeyOKH4Qa % eyOhIukapaqabaGcdaWcaaqaaerbfv3ySLgzaGqbc8qacqWFAbGwpa % WaaSbaaSqaa8qacqWFUbGBdaWgaaadbaGae83saSeabeaaaSWdaeqa % aOWdbmaabmaapaqaa8qacqWFtbWudaqadaWdaeaapeGaeqOSdiMaey % OeI0IaeqySdegacaGLOaGaayzkaaaacaGLOaGaayzkaaaapaqaa8qa % cqWGUbGBpaWaaSbaaSqaa8qacqWGRbWAa8aabeaaaaGcpeGaeyypa0 % ZaaSaaa8aabaWdbiabek7aIjabgkHiTiabeg7aHbWdaeaapeGaeGOm % aiJaeqiWdahaaaaa!571C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \;\;\mathop {\lim }\limits_{k \to \infty } \frac{{{Z_{{n_K}}}\left( {S\left( {\beta - \alpha } \right)} \right)}}{{{n_k}}} = \frac{{\beta - \alpha }}{{2\pi }}$$

(0.1)

for all sectors S(α, β).