- #36

Infrared

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julian said:We also have:

So write the polynomial as:

##

p(x) = a_n (x-a)^n + a_{n-1} (x-a)^{n-1} + \dots + a_4 (x-a)^4 + a_3 (x-a)^3 + a_0

##

where ##a_0 \not= 0##. We have an undulation point at ##x=a## if ##a_3 = 0## and the first non-zero term after it has even power in ##(x-a)##. Say this first non-zero term is ##a_{2m} (x-a)^{2m}##. For ##x## close to ##x=a## we have

##

p (x) \approx a_{2m} (x-a)^{2m} + a_0

##

so the corresponding trough or peak is not intersecting the ##x-##axis and again we must have complex roots?

Can you give a proof for why the fact that this approximation to [itex]p[/itex] having no real roots implies that [itex]p[/itex] has a corresponding non-real root?

Your count in the case that [itex]a[/itex] is an inflection point also confuses me. Can you give a more complete argument for why [itex]p(x)[/itex] has fewer than [itex]n[/itex] real roots.