On Sun, May 10, 2015 at 10:07 AM, Phoenix <[email protected]> wrote:
>
> The polynomial will be of integer coefficients always.
> k will be given as square-root of some integer.
>
> Does that help?

Like I said, you should provide a concrete example for whoever
actually tries to answer your question (probably not me).  Something
that a person can immediately use, e.g. a public sage worksheet or a
link to a sagecell would help even more: http://sagecell.sagemath.org/

William

>
>
>
> On Sunday, May 10, 2015 at 10:57:43 AM UTC-5, William wrote:
>>
>> 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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-- 
William (http://wstein.org)

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