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In 1D, it is straightforward to create real-valued coefficients polynomials which have no roots in $[0, 1]$: Pick a degree $n$ and some parameters $a_i$ outside of $[0, 1]$, and expand $$ p(x) = b \prod_{i=1}^n (x - a_i). $$ This isn't fully general though: Polynomials like $x^2+1$ are missed.

Is there a canonical form of real-valued polynomials with no zeros in $[0, 1]$?

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  • $\begingroup$ Doesn't your 1D form miss polynomials like $x^2+1$? $\endgroup$
    – user856
    Mar 21, 2018 at 14:46
  • $\begingroup$ @Rahul True. I'll guess I'll reformulate w.r.t. to 1D then. $\endgroup$ Mar 21, 2018 at 14:48

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The fully general form is $$ \alpha \prod_i L_i(x) \prod_j Q_j(x) $$ where $L$ is a monic linear polynomial without roots in $[0,1]$ and $Q_j$ is a monic quadratic polynomial without real roots. Each product may be empty. In other words, $$ \alpha \prod_i (x-a_i) \prod_j (x^2+b_jx+c_j) $$ where $\alpha, a_i, b_j,c_j \in \mathbb R$ with $\alpha\ne0$, $a_i \notin [0,1]$ and $b_j^2-4c_j <0$.

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