# Characterization of polynomials which have no zeros in the unit interval

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 $$[0, 1]$$?

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

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 $b_j^2-4c_j <0$.