Can someone explain how to express a polynomial in terms of elementary symmetric polynomial. (where there is possible) ? For example if i have the polynomials:
$t_1^3+t_2^3$
$t_1t_2^2+t_2t_3^2+t_3t_1^2$
$t_1^2+t_2^2+t_3^3$
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3Only symmetric polynomials can be expressed in terms of combinations of symmetric polynomials. Also, the first polynomial isn't symmetric. – Rushabh Mehta Mar 14 '20 at 20:09
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Don Thousand yes you mean : $t_1t_2^2+t_2t_3^2+t_3t_1^3$ right ,the 2nd one? – Lam18373 Mar 14 '20 at 20:18
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@Don Thousand can u explain how to express a symmetric polynomial in terms of elementary symmetric polynomials?? – Lam18373 Mar 14 '20 at 20:29
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@rogerl no cuz the answers there connected yo computer science not math! – Lam18373 Mar 14 '20 at 20:41
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@Almaa Okay, how about this one? – Viktor Vaughn Mar 15 '20 at 02:35
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Take your first polynomial for instance, which is symmetric in two variables: $x^3 + y^3$. We know that the ring of symmetric polynomials in two variables is generated by $e_1 = x + y$ and $e_2 = xy$, and we have $$ e_1^3 = x^3 + 3 x^2 y + 3 x y^2 + y^3 = (x^3 + y^3) + 3 xy (x + y) = (x^3 + y^3) + 3 e_2 e_1,$$ and hence $x^3 + y^3 = e_1^3 - 3 e_1 e_2$.
You can do the same ad-hoc method in three variables and you'll quickly find an answer. If you want to see more sophisticated ways of doing this, consult a book on symmetric function theory.
Joppy
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