91Ó°ÊÓ

Consider a polynomial \(p \in P^{n},\) i.e., \(p(z)=\prod_{k=1}^{n}\left(z-z_{k}\right)\) for some \(z_{k} \in \mathbb{C}\) (a) Write \(\log |p(z)|\) as a sum of \(n\) terms corresponding to the points \(z_{k}\) (b) Explain why the term involving \(z_{k}\) can be interpreted as the potential corresponding to a negative unit point charge located at \(z_{k},\) if charges repel in inverse proportion to their separation. Thus \(\log |p(z)|\) can be viewed as the potential at \(z\) induced by \(n\) point charges. (c) Replacing each charge -1 by \(-1 / n\) and taking the limit \(n \rightarrow \infty,\) we get a continuous charge density distribution \(\mu(\zeta)\) with integral \(-1,\) which we can expect to be related to the limiting density of zeros of polynomials \(p \in P^{n}\) as \(n \rightarrow \infty .\) Write an integral representing the potential \(\varphi(z)\) corresponding to \(\mu(\zeta),\) and explain its connection to \(|p(z)|\) (d) Let \(S\) be a closed, bounded subset of \(\mathbb{C}\) with no isolated points. Suppose we seek a distribution \(\mu(z)\) with support in \(S\) that minimizes \(\max _{z \in S} \varphi(z) .\) Give an argument (not rigorous) for why such a \(\mu(z)\) should satisfy \(\varphi(z)=\) constant throughout \(S\). Explain why this means that the "charges" are in equilibrium, experiencing no net forces. In other words, \(S\) is like a 2 D electrical conductor on which a quantity -1 of charge has distributed itself freely. Except for an additive constant, \(\varphi(z)\) is the Green's function for \(S\) (e) As a step toward explaining the rule of thumb of \(\mathrm{p} .279\), suppose that \(A\) is a real symmetric matrix with spectrum densely distributed in \([a, b] \cup\\{c\\} \cup[d, e]\) for \(a