How do I prove no two polynomials from a set of polynomials share a maxima between 0 and 1?

Asked Dec 09 '23 at 13:03

Active Dec 09 '23 at 17:32

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During my research I came across a problem regarding a lemma I needed to prove in order for an important result to hold later. The question is:

From the set of polynomials $$\{(1-x)(b_0 x + b_1 x^2 + \cdots + b_{n-1} x^n) | n \in \mathbb{N}\}$$ where each $b_i$ is either $0$ or $1$ but at least one of them needs to be 1. Do they all have a unique maxima on the x-axis between $0$ and $1$? ($0$ and $1$ because it is in the context of markov decision processes)

I tried to construct the corresponding set of derivatives and factor the polynomials from that set to find shared factors between pairs of polynomials in order to find shared roots, but I have still not found any general approach to this. Any help or tips would be appreciated a lot.

edited Dec 09 '23 at 17:32

asked Dec 09 '23 at 13:03

pixos12

1

Is $n$ a fixed integer? – Paul Frost Dec 09 '23 at 16:21
well, if all $b_i$ are zero, the resulting polynomial has an infinite number of maxima... – Exodd Dec 09 '23 at 16:34
@Exodd Sorry I forgot to add that at least one of them needs to be 1. – pixos12 Dec 09 '23 at 17:57
@PaulFrost No it's an arbitrary integer, I forgot to add that sorry. – pixos12 Dec 09 '23 at 17:58
If I understand it correctly, you ask whether each polynomial of the form $(1-x)(b_0 x + b_1 x^2 + \cdots + b_{n-1} x^n)$ has a unique maximum between $0$ and $1$. The set of all these polynomials is not an object which is needed here. – Kritiker der Elche Dec 23 '23 at 07:27

How do I prove no two polynomials from a set of polynomials share a maxima between 0 and 1?

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