From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3186 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: WHY ARE WE CONCERNED? I Date: Thu, 30 Mar 2006 10:03:57 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019142 7613 80.91.229.2 (29 Apr 2009 15:32:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:22 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Mar 30 18:52:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:52:10 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP5wD-0006P9-Im for categories-list@mta.ca; Thu, 30 Mar 2006 18:48:21 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 132 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:3186 Archived-At: On Wed, 29 Mar 2006, Vaughan Pratt wrote: > a few decades ago an elementary exposition of the Fundamental > Theorem of Algebra would not be expected to include an elementary proof > since the extant proofs were either lengthy arguments or nonelementary > appeals to the minimum modulus principle, properties of holomorphic > functions such as Liouville's theorem, or other results the reader would > be unlikely to be on top of. The dominant belief was that the only > short proofs were nonelementary ones. > > But for an audience aware only that z^i for any nonnegative integer i > maps circles at the origin to i-fold circles of radius r^i at the > origin, an entirely elementary notion, an expositor today would be > morally obligated to include a full proof since there is hardly anything > left to explain. The polynomial a_d z^d + ... + a_0 maps little circles > to the neighborhood of a_0 and big circles to a loop tending to a very > big d-fold circle of radius a_d r^d, whence the smoothly growing image, > under the polynomial, of a smoothly growing circle is obliged to cross > the origin at some stage. Still a topological argument, but now an > entirely elementary one. > > Except, that is, for the theorem that a loop wound d times around the > hole in the punctured plane cannot be continuously retracted to a point, > which was tacitly smuggled in there. But that statement is less > intimidating than anything based on holomorphic functions. > > This slick proof seems only to have emerged in the past couple of > decades. Has it? It seems to me no more than (an explicity homotopy-theoretic formulation of) the (implicitly homotopy-theoretic) proof via Rouch\'e's Theorem, which I was taught as an undergraduate (and which I've taught to undergraduates on many occasions since then). Peter Johnstone