From: Dario Bressanini Subject: Very nice and useful site! (...and a math question...) Date: Fri, 20 Aug 1999 16:39:38 +0200 (MET DST) Newsgroups: [missing] To: rusin@math.niu.edu Keywords: analytic algebraic geometry Hi, My name is Dario Bressanini (I am a quantum chemist) and first of all let me tell you that your site is very nice and very helpful for people like me that are not mathematicians but use math in their work. In your site i saw many answers to questions posed by many people, and i was wondering if you could help me, or maybe tell me where to look for an answer. The problem is the following: i usually work with real functions defined in a many dimensional space (R^N where N can be from 2 to say 100) and differentiable almost everywhere. The functions are NOT polynomials, but are built with standard functions: Exp, polynomials, Log.... Of particular interest are the hypersurfaces obtained by setting the functions to zero f(x,y,z,.....) = 0 Usually a function belongs to a given symmetry group so i know that has both positive and negative parts so it must have zeros. 1) Is there a way to tell in advance, or with minimal calculations, how many of those hypersurfaces there are? or better how many disjoint regions or R^N divided by those surfaces there are? 2) Is there a way to approximate arbitrarily well those hypersurfaces USING ONLY POLYNOMIALS? Let me give a simple example, consider f(x,y) = 0 (2-x) Exp[-x-2y]+(2-y) Exp[-y-2x] = 0 if I plot it i see that a crude approximation would be a circle a(x^2+y^2) with some a Is there a systematic way to improve the approximate polynomial description? (I tried with a taylor expansion but it does not work very well) What is the field that study this kind of problems? where should i look? Many Thanks Dario Bressanini ============================================================================== From: Dave Rusin Subject: Re: Very nice and useful site! (...and a math question...) Date: Fri, 20 Aug 1999 11:32:10 -0500 (CDT) Newsgroups: [missing] To: dario@fis.unico.it Thank you for your comments about my site. You seem to be just the kind of person whom I am trying to serve. Your question sounds interesting. My first guess is that this would be studied by people who work in several complex variables (that's MSC section 32) where the general theme is to study questions like those in algebraic geometry but where the sets are defined by analytic functions more general than just polynomials. I'm not particularly well-versed in this area but in particular cases I can perhaps say something useful. Let us look at the example you proposed: > (2-x) Exp[-x-2y]+(2-y) Exp[-y-2x] = 0 describes a curve in the plane which you wish to study. There are two parts of the question you might be interested in and the answers are different. On the one hand, you might want some global information about the structure of the curve; this is likely to be different from what one would see in algebraic geometry because there are no polynomials whose large-scale behaviour resembles the exponential function. On the other hand, you could about the local behaviour at any point; as you have guessed, this information is deduced from the taylor polynomials in general (you have to specify the points of interest around which the taylor series will be computed, of course). Your equation is simple enough to discuss both aspects. I separated the polynomial and exponential parts of the equation and discovered some simplification; in retrospect, it's easier to make the substitution {x=u+v,y=u-v} (which will be a 45-degree rotation and a simple scaling) which eventually lets us describe the curve as the graph of a function: u = (2-v+2*exp(-2*v)+exp(-2*v)*v)/(1+exp(-2*v)) So, far from the origin, we will either have v -> oo (where the curve is asymptotic to the line u = 2-v ) or v -> -oo (where it's asymptotic to the line u = 2+v). In the original coordinates, these asyptotes are the lines x=2 and y=2 respectively. As for local information, you'd have to specify the points of particular interest, but it seems to me the interesting part of the original curve is the part near the origin. (You described the curve as "like a circle" but I don't see that; I might describe it as a hyperbola or a parabola near the origin, but in some sense all the conic sections are "the same"...) Now, how exactly would you like to approximate this curve with polynomials? In this case I can use the Taylor series for my function of v : 2 4 5 u = 2 - v + 1/3 v + O(v ) to give approximations to any order. In a more general setting, you couldn't expect to solve for one variable as a function of the other, but you could still use Taylor series to approximate the _defining_ functions (e.g. your f with the Exp terms) by polynomials; it's then true that the surfaces defined by the polynomial equations approach the analytic surface. Now, there's another sense in which you might want to analyze the curve: you may not be interested in the infinitesimal region around x=2,y=2, nor about the behaviour as x,y -> +- oo, but rather something in the middle: perhaps what you want is to know the "shape" of that part of the curve in the first quadrant, say. Here I'm not really sure how to make the goal precise enough to say anything useful. You can indeed use polynomials in several ways to approximate portions of a curve, and the approximations can be made very good in many senses; but you need to decide how you want to accomplish this. For example, you might ask for the circle which minimizes the absolute distance from the curve (first quadrant portion) to the circle; or you might ask for the hyperbolas which minimizes the area between your curve and the hyperbola; or ... You have decide what family of curves (or polynomials) you wish to select from, and you must decide on the metric you will use to decide whether one fits better than another. These are basic questions to be answered any time you try to fit a model to data, as is done in statistics. If you have a clear sense of how these questions might be answered, then we can use them to describe the "shape" of this portion of the curve. All of this will of course become more complicated when you allow more variables, more functions, and more types of functions; but I think this one example you proposed already indicates how you need to phrase the questions. Let me know if I can be of more help. dave ============================================================================== From: Dario Bressanini Subject: Re: Very nice and useful site! (...and a math question...) Date: Fri, 20 Aug 1999 22:09:35 +0200 (MET DST) Newsgroups: [missing] To: Dave Rusin On Fri, 20 Aug 1999, Dave Rusin wrote: > Thank you for your comments about my site. You seem to be just the kind > of person whom I am trying to serve. Well, your site is very useful to people like me who are trying to understand the difference between algebric geometry, differential geometry, algebric topology etc.... > Your question sounds interesting. My first guess is that this would be As i told you, I am a quantum chemist, and my field is full of mathematical questions waiting for an answer, but unfortunately mathematicians rarely cross our field, unlike physics > is the part near the origin. (You described the curve as "like a circle" > but I don't see that; I might describe it as a hyperbola or a parabola Sorry, i was oversimplifying, in reality my x and y stands for x1^2+x2^2 and y1^2+y2^2 so x and y are positive only, the first quadrant you mentioned. The problem is that when using many variables (and the real case I am studying now has 12) I cannot "plot" or "see" the surface. As I told you, even knowing in how many different part the space is divided by the hypersurface would be very useful, but i cannot plot a 12 dimensional function > perhaps what you want is to know the "shape" of that part of the curve Exactly, as a first step i would be satisfied with a qualitative answer, for example, Is the hypersurface closed or not? As for the polynomials approximation, I would like to directly approximate the surface, not the function and then set it to zero. this in practice does not work since I do not know where to expand the taylor series. Atylor series gives a local information, i would like a global information.. I can give you a concrete example if you wish thanks dario