Regarding formulas: In general, the set of equations can be reduced to a single variable by elimination, but the degree of the single variable equation is the product of the original degrees. So the best you can do genericly is two quadratics in two variables. I once tried to write this out by hand -- a "biquadratic" formula -- but couldn't get through all the algebra. The quartic formula has the cubic formula as a sub-expression, so writing everything out is pretty big. Maybe Macsyma could tackle it. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Victor S. Miller [victorsmiller@gmail.com] Sent: Monday, November 30, 2009 12:01 PM To: Dan Asimov; math-fun Subject: Re: [math-fun] Solving polynomial equations with roots, etc. As far as your last point that amounts to checking whether or not the system generates the unit ideal. You can do that with a Groebner basis calculation which is souped up linear algebra. Victor Sent from my iPhone On Nov 30, 2009, at 1:15 PM, Dan Asimov <dasimov@earthlink.net> wrote:
We know that there exists a formula that may use roots (and arithmetic operations) for solving polynomials for each degree through 4, and no higher.
I. What's known about finding such a formula for solving *several polynomial equations in several variables* ?
(The polynomials can be of different degrees in general, and the coefficients can be any complex numbers.)
II. Wait! Let's go back a step and ask when such a system of polynomial equations is even *known to have a solution*.
--Dan
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