Monotonic polynomial

[olast1]

Main Question or Discussion Point

What general constraints on the coefficients of a polynomial of degree n do I need to impose to guarantee that this polynomial is strictly increasing on [0,1]?

 

[hypermorphism]

Take a look at what its Taylor series does to its coefficients.

 

[olast1]

Thank you for your answer, but I am not sure I understand what you mean. Can you explain?

 

[Hurkyl]

In what class was this problem given?

 

[olast1]

Master Level.

 

[Hurkyl]

"Master level"?

You mean something like, say, a graduate algebra course?

 

[olast1]

Yes, it is a Math class which is part of the first year in the master program in economics. Although I am not sure how, I hope this helps.

 

[Hurkyl]

Because the first thing that sprung to my mind was Sturm's theorem, but that's not something I would expect to be used in, say, a Real Analysis course.

So, it wouldn't be fruitful to suggest trying to turn the problem into a zero-finding problem if Sturm's theorem wasn't something you'd be expected to use!

 

[olast1]

Thank you for your suggestion. Sturm's theorem is something I could use. However, I am not sure how I can use it to find constraints on the coefficients that guarantee the monotonicity of the polynomial over [0,1].

 

[Hurkyl]

Well, my hint is to try and transform the original question into a question about finding roots -- what characterizations do you know of strictly increasing functions?

 

[olast1]

I am not sure I understand what you are getting at.

I do not know if it helps or overlap with what you are saying but here is what I tried to do at this point: if P(x) is a polynomial of degree n, then its derivative P'(x) is a polynomial of degree n-1. Therefore, I have tried to parametrize a polynomial of degre n-1 to guarantee that it is strictly greater than 0 for any x between [0,1]. The parametrization I found is

P'(x)=prod(i=1,...n-1){x-1/(1-Bi)} with Bi>0 for any i=1,...n-2 and Bn-1=exp[b*prod(i=1,...n-2){1-Bi}] and b>0 which I believe guarantees that P'(x)>0.

Now however, I am having problems relating the coefficients of P'(x) to the parameters of p(x).

Thank you for your time. I truly appreciate your help.

 

[Hurkyl]

I am having problems relating the coefficients of P'(x) to the parameters of p(x).

D'oh, that should be the easy part! If

$$p(x) = \sum_{i = 0}^{n} a_i x^i$$

Then you should be able to directly take a derivative, to get a formula for the coefficients of p'(x) in terms of that of p(x).

 

[olast1]

Well, what I am having problems with is to find a formula to relate the (B1,...,Bn-1) in my equation of P'(x) to your (a1,...an).

 

[Hurkyl]

Well, what's the derivative of a_i x^i?

(and don't forget about the constant terms...)

 

[olast1]

Let me rephrase, what I am having a problem with at this point is to relate the (B1,...,Bn-1) in my definition of P'(x) with the (C1,...,Cn-1) if I write P'(x) in the usual manner

P'(x)=sum(i=1,...,n-1){Ci*x^i}

Then obviously I can easily relate the Ci to your Ai.

 

[Hurkyl]

Oh, I feel silly. Sorry 'bout that!

I guess I was still thinking about how the Sturm's theorem approach would work, since that uses the coefficients of the polynomial directly. (Maybe I'm thinking about something related to Sturm's theorem than Sturm's theorem itself -- I can never keep them all straight, but that keyword is enough for me to find it in my reference materials!)