Discrepancy Theorems via One-Sided Bounds for Potentials

Abstract

In Chapter 2 we obtained discrepancy estimates for the zero distribution of a polynomial p in connection with the equilibrium measure µ _L of a Jordan curve or arc L. The basic quantities involved have been the two terms

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS % baaSqaaiaadchacaGGSaGaamitaaqabaGccqGH9aqpdaWfqaqaaiGa % c2gacaGGHbGaaiiEaaWcbaGaamOEaiabgIGiolablkqiJcqabaGcca % WGvbWaaWbaaSqabeaacqaH8oqBdaWgaaadbaGaamitaaqabaWccqGH % sislcaWG2bWaaSbaaWqaaiaadchaaeqaaaaakmaabmaabaGaamOEaa % GaayjkaiaawMcaaaaa!4BA0! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\varepsilon _{p,L}} = \mathop {\max }\limits_{z \in \mathbb{C}} {U^{{\mu _L} - {v_p}}}\left( z \right)$$

and

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS % baaSqaaiaadchacaGGSaGaamitaaqabaGccqGH9aqpcqaH1oqzdaWg % aaWcbaGaamiCaiaacYcacaWGmbaabeaakiabgkHiTiaadwfadaahaa % WcbeqaaiabeY7aTnaaBaaameaacaWGmbaabeaaliabgkHiTiaadAha % daWgaaadbaGaamiCaaqabaaaaOWaaeWaaeaacaWG6bWaaSbaaSqaai % aaicdaaeqaaaGccaGLOaGaayzkaaaaaa!4ADD! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\delta _{p,L}} = {\varepsilon _{p,L}} - {U^{{\mu _L} - {v_p}}}\left( {{z_0}} \right)$$

where z ₀ ∈ int L is fixed if L is a curve. In Section 2.3 we have outlined that it is possible to restrict the essential quantities to the outer bounds

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS % baaSqaaiaadchacaGGSaGaamitaaqabaGcdaqadaqaaiaadkhaaiaa % wIcacaGLPaaacqGH9aqpdaWfqaqaaiGac2gacaGGHbGaaiiEaaWcba % GaamOEaiabgIGiolaadYeadaWgaaadbaGaamOCaaqabaaaleqaaOGa % amyvamaaCaaaleqabaGaeqiVd02aaSbaaWqaaiaadYeaaeqaaSGaey % OeI0IaamODamaaBaaameaacaWGWbaabeaaaaGcdaqadaqaaiaadQha % aiaawIcacaGLPaaacaGGSaGaaGjcVlaadkhacqWI7jIzcaaIWaaaaa!544F! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\varepsilon _{p,L}}\left( r \right) = \mathop {\max }\limits_{z \in {L_r}} {U^{{\mu _L} - {v_p}}}\left( z \right),{\kern 1pt} r \succ 0$$

in the case of a Jordan arc. If L is a curve, we replace δ _p,L by the smaller inner bound

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS % baaSqaaiaadchacaGGSaGaamitaaqabaGcdaqadaqaaiaadkhaaiaa % wIcacaGLPaaacqGH9aqpdaWfqaqaaiGac2gacaGGHbGaaiiEaaWcba % GaamOEaiabgIGiolaadYeadaqhaaadbaGaamOCaaqaaiabgkHiTaaa % aSqabaGccaWGvbWaaWbaaSqabeaacqaH8oqBdaWgaaadbaGaamitaa % qabaWccqGHsislcaWG2bWaaSbaaWqaaiaadchaaeqaaaaakmaabmaa % baGaamOEaaGaayjkaiaawMcaaiabgkHiTiaadwfadaahaaWcbeqaai % abeY7aTnaaBaaameaacaWGmbaabeaaliabgkHiTiaadAhadaWgaaad % baGaamiCaaqabaaaaOWaaeWaaeaacaWG6bWaaSbaaSqaaiaaicdaae % qaaaGccaGLOaGaayzkaaGaaiilaiaayIW7caaIWaGaeSOEIaNaamOC % aiablQNiWjaaigdaaaa!62DB! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\delta _{p,L}}\left( r \right) = \mathop {\max }\limits_{z \in L_r^ - } {U^{{\mu _L} - {v_p}}}\left( z \right) - {U^{{\mu _L} - {v_p}}}\left( {{z_0}} \right),{\kern 1pt} 0 \prec r \prec 1$$

where L ⁻ _r is the level line of the conformal mapping φ(z) of int L onto D normalized by φ(z ₀) = 0, φ'(z ₀) > 0, as in (1.4.5). Then the discrepancy estimates can be formulated in terms of ε _p,L (r) + δ _p,L (r). In this chapter we shall discuss this approach carefully for general signed measures.