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Given a polynomial $y : \mathbb{R} \mapsto \mathbb{R}$ of degree $p$: $$ y(x) = \sum_{k=0}^p c_k\, x^k,$$ can a random set of coefficients $\{c_0, \cdots ,c_p\}$ be generated such that $y$ is monotonically increasing for all $x \in [0,1]$?

Alternatively, can a probabilistic bound on the monotonicity of $y$ be shown? For example, if $n$ tests of the nonnegativity of the slope of $y$ are made at different locations $\{x_1,\cdots, x_n\}$, can an upper bound be placed on the probability that $y$ is not monotonically increasing?

Finally, if $\epsilon$ is this bound, then can $\epsilon$ be written as a function of $n$ and $p$?

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    Well, if you force your coefficients to be all greater than 0 (say, they each follow an exponential distribution), your polynom will be monotonically increasing on $[0,1]$, and even on $\mathbb{R}^+$.2012-07-18
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    I'm sure (almost) you don't mean the domain of $y$ to be $\mathbb{R}^p$. As a function $y(x)$ maps $\mathbb{R}$ to $\mathbb{R}$, and even if some dependence on coefficients $c_i$ is intended, there are $p+1$ of those. $p$ must be odd for $y(x)$ to be monotone on all of $\mathbb{R}$; it's essentially asking that the derivative $y'(x)$ be nonnegative. The reduction to interval $[0,1]$ is similar. As far as a random such choice goes, we'd need to specify the probability distribution desired.2012-07-18
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    If you are looking for a numerical technique, then you could select the coefficients 'randomly' and then check if the resulting polynomial is increasing (just check if the polynomial $y'$ is non-negative on $[0,1]$ by looking at the end points and real roots).2012-07-18
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    I've corrected the domain of $y$. I do have future plans to expand the problem to higher dimensions, but that was a typo.2012-07-19

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