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MathGroup Archive 2001

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Re: Sum of Squares

  • To: mathgroup at smc.vnet.net
  • Subject: [mg26611] Re: [mg26587] Sum of Squares
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Thu, 11 Jan 2001 10:39:19 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Yes, this is certainly  possible. For the case of quadratic polynomials in
several variables, by using the same methods as in the general
classification of real quadrics (see any good linear algebra book) one can
easily show that any such expression which is non-negative for all real
values of the variables can be expressed a sum of squares of linear
expressions. I am sure that is also true for higher order polynomial
expressions (i.e. that they must be sums of squares) although I can't
immediately see an obvious way to prove it.

It is should not be difficult to write a Mathematica package which does this
(a package that does this may well already exist).

By the way, it is easy to check using Mathemaitca if a polynomial expression
is always non-negative. For example in your case:

 In[1]:=
<< Algebra`InequalitySolve`
In[2]:=
InequalitySolve[5 x^2 + 8 x y + 5 y^2 + 2 x z - 2 y z + 2 z^2 >= 0, {x, y,
z}]
Out[2]=
True

-- 
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/


on 1/9/01 3:52 PM, Will Self at wself at msubillings.edu wrote:

> Is there, or could there be, a "sum of squares" function for
> multivariate polynomials, which would rewrite an expression as a sum of
> squares, when possible?
> 
> It might work like this:
> 
> SumOfSquares[5 x^2 + 8 x y + 5 y^2 + 2 x z - 2 y z + 2 z^2]
> 
> --->
> 
> (x + 2y - z)^2 + (2x + y + z)^2
> 
> Will Self
> email appreciated
> 
> 
> Sent via Deja.com
> http://www.deja.com/
> 
> 



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