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April 7th, 2009, 01:16 AM  #1 
Newbie Joined: Apr 2009 Posts: 19 Thanks: 0  Newton's method for multivariate polynomial equation systems
I'm looking to apply Newton's method to solve a system of polynomial equations, where all polynomials are of degree two. Since for a single polynomial equation of degree 2 Newton's method always converges (except for singular points), I was wondering whether a similar theorem holds for the multivariate case. Anyone knows? 
April 7th, 2009, 06:05 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Newton's method for multivariate polynomial equation systems
I would expect so, since polynomials are wellbehaved (in particular they have second derivatives). But I haven't seen a proof or even a claim of that.

April 7th, 2009, 04:34 PM  #3 
Newbie Joined: Apr 2009 Posts: 19 Thanks: 0  Re: Newton's method for multivariate polynomial equation systems
Actually, after thinking about it some more, it shouldn't work in general for degree two  what you need should be some convexproperty I reckon. Reason why degree 2 doesn't mean much: you can emulate any polynomial by a set of polynomial equations of degree 2, e.g. x^3  x^2 + 1 = 0 (Newton can fail here I believe) is equivalent to x^2  y = 0 xy  y + 1 = 0 Any ideas about theorems involving convex functions? 

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equation, method, multivariate, newton, polynomial, systems 
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