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.KeywordsConformal MappingJordan CurveDiscrepancy TheoremAnalytic Jordan CurveOuter BoundThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.