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Let $$P_{t}(z) =a_{0}(t) + a_{1}(t)z + ...+a_{n}(t)z^n$$ be a polynomial where the coefficients depend continuously on a parameter $t \in (−1, 1)$.

Assume that there exists $\text{t}_{0} \in (−1, 1)$ such that $\text{t}_{0} \neq 0$ and the roots of $P_{t}(z)$ are distinct (no multiple roots). Show that the same is true for $P_{t}$ when $t$ is sufficiently close to $\text{t}_{0}$.

I am thinking of applying Riesz representation lemma, but I am still looking for an expanded form solution.

Samrat Mukhopadhyay
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user145993
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  • http://meta.matheducators.stackexchange.com/questions/93/mathjax-basic-tutorial-and-quick-reference – Timbuc Dec 12 '14 at 05:14
  • There was an almost identical question (suggesting the use of Rouché's theorem instead of Riesz's) asked a couple of days ago, but I can't find it... Anyway, you don't need any fancy theorem, just use the continuity of the coefficients. – hjhjhj57 Dec 12 '14 at 05:28
  • This looks highly relevant: http://math.stackexchange.com/questions/63196/continuity-of-the-roots-of-a-polynomial-in-terms-of-its-coefficients

    One of the answers uses an argument very close to the one used in Rouché's theorem proof, as was suggested in the other post.

     – hjhjhj57 Dec 12 '14 at 09:25
  • Thank you for this link it didn't help me. Do you have any idea? – user145993 Dec 12 '14 at 15:55
  • not even Joel Cohen's answer? – hjhjhj57 Dec 12 '14 at 20:00

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