An application of newtons sum

For a given polynomial of degree nn with roots r1,r2,rn r_1, r_2, \ldots r_n , let sks_k be the kth symmetric sum and pk=i=1nrikp_k = \sum_{i=1}^n r_i ^k is the kth power sum

Claim: If s1=s2=s3=...=sk=0s_1=s_2=s_3=...=s_k=0,then p1=p2=p3=.,..=pk=0p_1=p_2=p_3=.,..=p_k=0. where


Proof: By induction on kk . Base case: Since s1=p1 s_1 = p_1 , hence if s1=0 s_1 = 0 then p1=0 p_1 = 0 .

Induction step: Suppose that for some jj, we know that p1,p2,pj=0 p_1, p_2, \ldots p_j = 0 . Then, by newtons sum, we have

pk=s1pk1s2pk2+....+(1)k1sk2p2+(1)ksk1p1+(1)k+1ksk=0+0++0.p_k=s_1p_{k-1}-s_2p_{k-2}+....+(-1)^{k-1}s_{k-2}p_2+(-1)^{k}s_{k-1}p_1+(-1)^{k+1}ks_k = 0 + 0 + \ldots + 0.

Note by Aareyan Manzoor
3 years, 8 months ago

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There's no need to apply Newton's Sum.

If s1=s2==sk=0s_1 = s_2 = \cdots = s_k = 0 , then the roots satisfy the equation, xk=0x^k = 0 . So all its roots are 0, then by definition, p1=p2==pk=0p_1 = p_2 = \cdots = p_k = 0 .

Pi Han Goh - 3 years, 8 months ago

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what about x100x2+x+1=0x^{100}-x^2+x+1=0 i meant for that

Aareyan Manzoor - 3 years, 8 months ago

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I don't follow. What about that polynomial?

Pi Han Goh - 3 years, 8 months ago

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@Pi Han Goh look at this

Aareyan Manzoor - 3 years, 8 months ago

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@Aareyan Manzoor That one only applies for s1=s2==s98=0s_1 = s_2 = \ldots = s_{98}= 0 . And s99=990s_{99} = -99 \ne 0 .

Pi Han Goh - 3 years, 8 months ago

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@Pi Han Goh i meant that see, both variables are "n".

Aareyan Manzoor - 3 years, 8 months ago

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@Aareyan Manzoor I don't understand what you're saying.

Pi Han Goh - 3 years, 8 months ago

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@Pi Han Goh i gave wrong variables in note, just realized now. edited

Aareyan Manzoor - 3 years, 8 months ago

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@Aareyan Manzoor Great note! I've cleaned it up so that it's much easier to understand what you are saying.

Presenting it via a proof by induction isn't necessary, but helps build the idea that all we need to show pj=0 p_j = 0 is up to sj=0 s_j = 0 .

By improving the format of your presentation, you make it immediately clear what you want to show, and why it's interesting.

Calvin Lin Staff - 3 years, 8 months ago

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thank you sir!

Aareyan Manzoor - 3 years, 8 months ago

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nice way....

Dev Sharma - 3 years, 8 months ago

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