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Re: Sum of Products - Complete The Square

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126714] Re: Sum of Products - Complete The Square
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Sat, 2 Jun 2012 05:41:48 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201205310651.CAA27227@smc.vnet.net>


On 31 May 2012, at 08:51, djmpark wrote:

> Thanks for introducing/reminding us of SymmetricPolynomial and the
> associated SymmetricReduction. SymmetricReduction is the high tech
> generalization of complete the square.
>
> 3 x^2 + y^2 + 6 x y;
> Total@SymmetricReduction[%, {x, y}]
>
> 2 y^2 + 3 (x + y)^2

1. This is not quite right:

3 x^2 + y^2 + 6 x y;
Total@SymmetricReduction[%, {x, y}]

 -2 y^2 + 3 (x + y)^2

2. I don't see why this a "generalisation of complete the square". In 
fact, there is no well defined notion of completing the square for 
arbitrary quadratic expressions in two variables, e.g. this is another 
one:

(1/3)*(3*x + Sqrt[3]*y)^2 + x*y*(6 - 2*Sqrt[3])

In any case, symmetric reduction is about symmetric polynomials and not 
about squares and it is based not on anything "high tech" but in fact on 
an extremely classical fact, (known as the fundamental theorem of 
symmetric functions: 
http://mathworld.wolfram.com/FundamentalTheoremofSymmetricFunctions.html) 
that gave rise to the subject of invariant theory, namely, that the ring 
of invariants of the symmetric group on n symbols acting on polynomials 
in n-variables is a polynomial ring in n-variables. These new variables 
are known as the elementary symmetric functions.

The seeming relation with completing the square is quite incidental. The 
only reason why it looks this way is that the first elementary 
polynomial is the sum of the variables so if the expression is a 
quadratic you will get in it (x+y+=85)^2. But:

Total[SymmetricReduction[(2 x + y)^2, {x, y}]]

-4 x y - 3 y^2 + 4 (x + y)^2

A complete square was turned in to an incomplete one. One would hardly 
want to call this "completing the square" would one?

Andrzej Kozlowski









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