No. There's an answer to this question, and the computer is the wrong way to do it.
Let me lay out some assumptions and we can all have a good laugh about it later. *Remember when you said "I have the answer" and instead you had drew a picture of a pony?* And I'll say *no, it was a unicorn.* But for now let me pretend like I know what I'm talking about. I need a working idea of the problem. 1) The computer is the wrong way to go about it. We have brains and computers don't. I want a solution to a quadratic system. I want a quadratic solution. The assumption that, say 'an arrangement of terms that will 'do' for a first-order system is a well-formed question' is unfounded. Perhaps, obviously untrue. 2) I need (or would like) a finite number of rational points to *define* a solution space by a new, more elegantly chosen set of quadratic inequalities. Namely... ellipses. 3) I have to make a *series of choices* about the solution space and the terms before me. They can be arranged arbitrarily, in the algebra, but that does not necessarily describe the system in question. I need to investigate what I mean by the pairings of terms. Are they really just indistinguishable pairs-among-equals? If this is *really true* then I should just say to hell with it and wrap each distinct pair up as a separate variable, and write it as a linear system in n/2 independent variables. 4) But suppose we don't really believe it's just a mess of linear equations that accidentally got tangled up. Say I want to keep the meaning of the quadratic form, but I *do not *know enough about the system to preference any one arrangement of the terms over another. 5) this is a convenient way to "define" the general case of the problem. It's a bit of a lie. I could fool myself into thinking I had the THE correct answer for a particular problem when in reality what I mean is that I have used, say, the minimim number of rational points necessary to define the solution space. There is no reason to assume such a minimization is meaningful in any particular case. Brushing that aside, however, and embracing abstraction, 6) I want to throw lassos around the minimum elements of the solution space. I want to rewrite the system without "solving" the quadratics, but by *first* arranging it into a series of ellipses which I will say * define* different overlapping regions. Then I have bounded the solution space in the most flexible and economic geometric way. I think. I mean. It seems like it. The ellipse seems like a magical little tool. It defines a *fully bounded* 2-d section of space, and simultaneously describes the relative scale of its geometric proportion within the plane chosen. Then I go hunting for how to slide those ellipses along the cones, among the various dimensions, until O can get them all to share points. The solution can be *unlatched*. We have to hunt it, and make it give its cookies away. Now, suppose we have enough dimensions that there is no necessary way to do this? That fact, by itself contributes even more descriptive information about our system. Information crucial to understanding its structure, and which we will lack, holding a computer printout to a system of equations. I am not sure how to proceed from here but in the worst-case scenario maybe we could define a klunky, parametric form that forces us to make some assumptions about how the terms fit together, and start hunting for asymptotes. ?? I don't think I'll make it that far before having to revise everything I just wrote. I need to figure out just how quickly the number of possible conic arrangements expands as the system increases in size. I am making the assumption that it will always be possible to sort out possible asymptotes, brute force some boundary points if we have to, but there is no necessary reason for that to be true. In any case, I think the human manipulation of the terms into a solvable form is really what this is about. Thoughts? And. please. Seriously. I need to know if I'm reasoning like a child. -- To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
