# Monotonic polynomial

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]?

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

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

Hurkyl
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Master Level.

Hurkyl
Staff Emeritus
Gold Member
"Master level"?

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

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
Staff Emeritus
Gold Member
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! 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
Staff Emeritus
Gold Member
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?

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).

Hurkyl
Staff Emeritus
Gold Member
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).

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
Staff Emeritus
Gold Member
Well, what's the derivative of ai x^i?

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

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
Staff Emeritus