I have two polynomials p(x) and q(x) and I want to know if there are roots of the equation p′p=q′q in the domain (a,∞) , where a=maxroots(p),roots(q)
This is the same as asking for the roots of the polynomial, p′q−pq′=0 in the same domain.
I would do as follows:
sage: r = p.derivative()*q - q*p.derivative()
sage: R_r = r.roots(RR)This gives you the real roots of r in the variable R_r. Now to filter the roots that make sense to you, you need to compute the roots of p and q:
sage: R_p = p.roots(RR)
sage: R_q = q.roots(RR)In R_p and R_q, you have the real roots of p and q, given as pairs with the root and the multiplicity. To find a:
sage: a = max([x[0] for x in R_p] + [x[0] for x in R_q])And finally, the roots you want:
sage: R = [x in R_r if x[0] > a]Thanks! Let me try this.
If you just wrote "p.roots()" would it give all the roots? (and not just the reals)
And what would be a command to check if R turned out to be empty? I need some kind of an error catching for that.
If I am reading the code correctly and R is a list, then you could try len(R) == 0 . If I misunderstood, then nevermind.
Alternatively
def are_there_roots(p, q):
return FalseWhat if, for example, p=q≠0 or if p′ and q′ have a common root that is larger than a ?
I think it's hard for p′ and q′ to have a common root larger that the largest root of p and q... I am impressed by how exact yet uninformative this answer is!
@b-r-u-n-o wrote:
I think it's hard f=or p′ and q′ to have a common root larger that the largest root of p and q...
What about, say, p=q=x3+8 whose single real root is -2, but whose derivative have root 0 ?
max(roots(p), roots(q)) is ambiguous.
Asked: 9 years ago
Seen: 578 times
Last updated: Jun 12 '15
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