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Re: Sum of Squares
*To*: mathgroup at smc.vnet.net
*Subject*: [mg26625] Re: [mg26587] Sum of Squares
*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>
*Date*: Sat, 13 Jan 2001 22:36:06 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
A small correction is needed to my earlier note. It is indeed true that
every polynomial of degree two is a sum of squares of polynomials and this
indeed follows from the diagonalization of quadratic forms (or
classification of real quadrics) as I wrote. I was wrong in believing that
this was also true for polynomials of arbitrary degree. What is true is
that every such polynomial is a sum of squares of rational functions. This
is the solution of "Hilbert's 17th problem". An example of a polynomial
which is always non-negative but is not a sum of squares of polynomials is:
w^4+x^2*y^2 +y^2*z^2 +z^2*x^2 -4x y z w
Hilbert's 17 thh problem and related questions are considered in detail in
Bochnak, Coste and Roy: "Real Algebraic Geometry", ch. 6.
on 01.1.10 9:52 AM, Andrzej Kozlowski at andrzej at tuins.ac.jp wrote:
> 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/
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