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I was browsing through the Art Of Problem Solving website and came across this:

"Any symmetric sum can be written as a Polynomial of the elementary symmetric sum functions, for example

$x^3 + y^3 + z^3 = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = S^3_1 - 3S_1S_2 + 3S_3$ (*)

This is often used to solve systems of equations involving power sums, combined with Vieta's formulas."

However, I do not understand how the expression (*) above was derived and, I also do not understand how it can be used to solve systems of equations involving power sums. Please someone should help with a detailed explanation and also an example on how it can be used to solve systems of equations involving power sums.

RobPratt
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Icosahedron
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1 Answers1

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There is a simple algorithm of Gauss to rewrite a symmetric polynomial $f(x,y)$ as polynomial in the elementary symmetric polynomials $\,s_1\! = x+y,\ \ s_2 = xy.\,$ Namely if $\,f\,$ has highest degree term $\ c x^a y^b $ in the lex (dictionary) order (i.e. $\,(a,b) > (c,d)\, $ if $\,a >c,\,$ or $\,a= c\,$ and $\, b > d)\,$ then cancel the highest term of $\,f\,$ by subtracting $\,cs_1^{a-b} s_2^b,\, $ then recurse on what remains.

Let's perform Gauss's algorithm on the simpler example $\, f = x^3 + y^3.\, $ Since $\,(3,0) > (0,3)\,$ the highest degree monomial is $\ 1\cdot x^\color{#0a0}3y^\color{#c00}0,\, $ so we subtract $\ 1\cdot s_1^{\color{#0a0}3-\color{#c00}0} s_2^\color{#c00}0\, =\, (x+y)^3 $ yielding

$$\ x^3+y^3\ -\ (x+y)^3\, =\ {-}3x^2 y - 3x y^2$$

By $(2,1)>(1,2),\, $ RHS has high term $\,-3x^{\color{#0a0}2} y^\color{#c00}1$ so we subtract $\, {-}3 s_1^{\color{#0a0}2-\color{#c00}1} s_2^\color{#c00}1 =\, -3(x\!+\!y)(xy)$

$$\ x^3+y^3\, -\ (x+y)^3\, +\ 3(x+y)(xy) \ =\ 0$$

So the algorithm terminates, yielding $\ f = s_1^3 - 3s_1 s_2.\ $

Exactly the same algorithm works for polynomials in any number of variables: for the high term $\,c\, x_1^{\large a_1}\cdots x_k^{\large a_k}\,$ we subtract $\,c\, s_1 ^{\large a_1-a_2} s_2^{\large a_2-a_3}\cdots s_{k-1}^{\large a_{k-1}-a_k}s_k^{\large a_k},\,$ e.g. in your trivariate example, for high term $\,c\, x^a y^b z^c\, $ we subtract $\,c\, s_1^{a-b} s_2^{b-c} s_3^c.\,$ This reduces such problems to rote mechanical computation, i.e. no guesswork is required to solve such problems, only simple polynomial arithmetic. The algorithm yields a constructive interpretation of the Fundamental Theorem of Symmetric Polynomials, that every symmetric polynomial has a unique representation as a polynomial in the elementary symmetric polynomials.

Gauss's algorithm may be viewed as a special case of Gröbner basis methods (which may be viewed both as a multivariate generalization of the (Euclidean) polynomial division algorithm, as well as a nonlinear generalization of Gaussian elimination for linear systems of equation). Gauss's algorithm is the earliest known use of such a lexicographic order for term-rewriting (now mechanized by the Grobner basis algorithm and related methods).

Bill Dubuque
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    I had no idea the algorithm was so simple. I am a geometer by training, but have had to express symmetric polynomials in terms of the elementary symmetric polynomials on more than one occasion. (They come up, for example, in the computation of cohomology rings of homogeneous spaces) Each time, I've worked them out by essentially a guess and check method. This is so nice! Thanks for posting! – Jason DeVito - on hiatus Aug 08 '17 at 16:42
  • Thank you so much! I've spend so much time trying to understand this algorithm, but now I do! – slowpoke Oct 29 '21 at 15:01