If you figure out that all roots are real, I'm pretty sure the best you can do about checking if k is larger than all roots, is to take the smallest of the bounds on roots here <https://en.wikipedia.org/wiki/Properties_of_polynomial_roots#Bounds_on_.28complex.29_polynomial_roots>, and checking if k is larger than that.
Note that the previous checks the absolute value, so if you don't care about negative roots, take a look at this paper <http://www.inf.uth.gr/wp-content/uploads/formidable/phd_thesis_vigklas.pdf> or this paper <http://www.jucs.org/jucs_15_3/linear_and_quadratic_complexity>. The things mentioned in those papers should be implemented in sage, but I don't know where. man. 11. maj 2015 kl. 07.43 skrev Phoenix <[email protected]>: > > Here is an example of a real-rooted polynomial generated by my code: > > 1296*x^24 - 20736*x^22 + 129600*x^20 - 393984*x^18 + 584496*x^16 - > 362880*x^14 + 62208*x^12 > > > This I know to be real-rooted because I tested this in Mathematica. > > > In general my code will generate polynomials of degree ~200-300 > > Then is there a way to test real-rootedness inside SAGE itseld? > > And given some other number (typically as a squareroot of some other integer) > can I test if the largest root is below that threshold? > > > > > On Sunday, May 10, 2015 at 3:09:48 PM UTC-5, vdelecroix wrote: >> >> On 10/05/15 21:51, Dima Pasechnik wrote: >> > On Sunday, 10 May 2015 19:11:27 UTC+1, vdelecroix wrote: >> >> On 10/05/15 19:43, Dima Pasechnik wrote: >> >>> A random polynomial for sure won't have the properties your >> polynomial >> >>> probably has (e.g. all roots real). >> >> >> >> Nope, but the characteristic polynomial of a "random" matrix in >> SL(d,R) >> >> would ;-) >> >> >> > >> > really? I'd say a random symmetric matrix in M_d(R), yes, surely, but >> not >> > in SL_d(R)... >> >> Depends on your definition of random. If you pick any non-degenerate >> generating set and look at a random product of length > d^2 then yes. >> >> Though, I do not know for the definition of random as "uniform on all >> matrices in SL_d(R) with coefficients less than N". >> >> Vincent >> > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
