My Math Forum Newton's method for multivariate polynomial equation systems

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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 well-behaved (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 convex-property 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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