On Sun, May 10, 2015 at 8:43 AM, Phoenix <[email protected]> wrote:
> I have a polynomial which is of degree ~500 and a number $k$.
> I want to check two properties,

Your question is far too ambiguous.   You should make available an
explicit example of a polynomial and a k that are want to try to do
this with.    For example, this problem is completely different
depending o whether the coefficients of the polynomial are floating
point or say huge exact rational numbers, or maybe numbers in an
algebraic number field...

>
> - That all the roots of this polynomial are real
>
> - That the largest root is below $k$.
>
> What is the fastest way to check this?
>
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-- 
William (http://wstein.org)

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