From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3711 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: functions not polynomials Date: Wed, 28 Mar 2007 20:47:27 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019476 9918 80.91.229.2 (29 Apr 2009 15:37:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:56 +0000 (UTC) To: categories list Original-X-From: rrosebru@mta.ca Thu Mar 29 23:28:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Mar 2007 23:28:10 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HX6he-0003cv-6X for categories-list@mta.ca; Thu, 29 Mar 2007 23:18:58 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 65 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:3711 Archived-At: Paul Taylor's example of an f(x,y) that is polynomial separately in x and y but not jointly was sum_n (x,n)(y,n) (where (x,n) denotes the binomial coefficient x!/(n!(x-n)!)). After mulling over that example some more it occurred to me that it can be analyzed via the observation that W^{-1} maps the polynomial P_n(x) = (x(x-1)(x-2)...(x-(n-1)))^2 to the polynomial xy(x-1)(y-1)(x-2)(y-2)...(x-(n-1))(y-(n-1)). This contradicts my earlier claim that the only polynomials in x that W^{-1} maps to polynomials in x and y are the linear combinations of 1 and x^2. These two are easily seen to be the only *monomials* so mapped, but (the linearity of W^{-1} notwithstanding) it does not follow that the only *polynomials* so mapped are the linear combinations of these two monomials. W^{-1} maps sum_n P_n(x)/(n!)^2 to Paul's example. The coefficient 1/(n!)^2 of P_n(x) seems to play no role here, and any coefficients should do as long as infinitely many are nonzero (to make f(x,y) not a polynomial). To extend the example (as a function on N^2) directly (via the constituent polynomials) to a function on the positive reals however, the coefficients would need to grow somewhat slower than 4^n, |P_n(x)| being bounded above by at best about 1/4^n for 0 < x < n (the half-integer points for x in that range give a good approximation of the bound). 1/(n!)^2 is more than slow enough for this purpose. A simpler example is g(x) = (2x,x) (again the binomial coefficient), which W^{-1} maps to f(x,y) = (x+y,x), polynomial separately in x and y but not jointly. That is, Pascal's triangle is a sufficient counterexample for Paul's purposes. Moreover the Gamma function gives a nicer (log-convex in fact) extension of f(x,y) to the upper right quadrant of R^2. Vaughan Pratt