“Reverse” Vieta's Formulas












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In one of my investigations, I needed to figure out sums of powers of roots of polynomial equations. These are not very hard to figure out. For monic polynomials, the first three sums are:



Sum of roots: -a_{n-1}

Sum of squares: a_{n-1}²-2a_{n-2}

Sum of cubes: -a_{n-1}³+3a_{n-1}a_{n-2}-3a_{n-3}


Would anyone have a reference where all of these formulas can be found? Do you know of any applications that these formulas have?






polynomials roots symmetric-polynomials







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  • 1




    $begingroup$
    en.wikipedia.org/wiki/…
    $endgroup$
    – Qiaochu Yuan
    Dec 4 '18 at 1:24






  • 1




    $begingroup$
    I think you'll find what you need in this
    $endgroup$
    – EvanHehehe
    Dec 4 '18 at 1:24
















1












$begingroup$


In one of my investigations, I needed to figure out sums of powers of roots of polynomial equations. These are not very hard to figure out. For monic polynomials, the first three sums are:



Sum of roots: -a_{n-1}

Sum of squares: a_{n-1}²-2a_{n-2}

Sum of cubes: -a_{n-1}³+3a_{n-1}a_{n-2}-3a_{n-3}


Would anyone have a reference where all of these formulas can be found? Do you know of any applications that these formulas have?






polynomials roots symmetric-polynomials







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/…
    $endgroup$
    – Qiaochu Yuan
    Dec 4 '18 at 1:24






  • 1




    $begingroup$
    I think you'll find what you need in this
    $endgroup$
    – EvanHehehe
    Dec 4 '18 at 1:24














1












1








1





$begingroup$


In one of my investigations, I needed to figure out sums of powers of roots of polynomial equations. These are not very hard to figure out. For monic polynomials, the first three sums are:



Sum of roots: -a_{n-1}

Sum of squares: a_{n-1}²-2a_{n-2}

Sum of cubes: -a_{n-1}³+3a_{n-1}a_{n-2}-3a_{n-3}


Would anyone have a reference where all of these formulas can be found? Do you know of any applications that these formulas have?






polynomials roots symmetric-polynomials







share|cite|improve this question









$endgroup$




In one of my investigations, I needed to figure out sums of powers of roots of polynomial equations. These are not very hard to figure out. For monic polynomials, the first three sums are:



Sum of roots: -a_{n-1}

Sum of squares: a_{n-1}²-2a_{n-2}

Sum of cubes: -a_{n-1}³+3a_{n-1}a_{n-2}-3a_{n-3}


Would anyone have a reference where all of these formulas can be found? Do you know of any applications that these formulas have?






polynomials roots symmetric-polynomials




polynomials roots symmetric-polynomials






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 4 '18 at 1:17









LemmaLemma

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63529








  • 1




    $begingroup$
    en.wikipedia.org/wiki/…
    $endgroup$
    – Qiaochu Yuan
    Dec 4 '18 at 1:24






  • 1




    $begingroup$
    I think you'll find what you need in this
    $endgroup$
    – EvanHehehe
    Dec 4 '18 at 1:24














  • 1




    $begingroup$
    en.wikipedia.org/wiki/…
    $endgroup$
    – Qiaochu Yuan
    Dec 4 '18 at 1:24






  • 1




    $begingroup$
    I think you'll find what you need in this
    $endgroup$
    – EvanHehehe
    Dec 4 '18 at 1:24








1




1




$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 1:24




$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 1:24




1




1




$begingroup$
I think you'll find what you need in this
$endgroup$
– EvanHehehe
Dec 4 '18 at 1:24




$begingroup$
I think you'll find what you need in this
$endgroup$
– EvanHehehe
Dec 4 '18 at 1:24










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