# Parameterize a polynomial with no real roots

An even polynomial with a constant term of 1 will have no real roots if the coefficients of the powers (the c's below) are non-negative. So

$$1 + c_2x^2 + c_4x^4 + c_6x^6$$

has no real roots. Is there a general way to parameterize an nth order polynomial with a constant term of 1 so that it has no real roots? I know that the above conditions (even powers, with non-negative coefficients) are more restrictive than necessary. The application is fitting (x,y) data where y is always positive with a polynomial in x.

## 1 Answer

Let $p(x)=1+c_1+\dots c_n$. Since $p(0)=1>0$, if $p$ does not have real roots, it must be positive. This implies $c_n>0$ (otherwise there would be a positive root) and $n$ even (otherwise there would be a negative root.) Applying Descartes rule of signs to $p(x)$ and $p(-x)$ we get the following necessary condition: the sequences of coefficients $$1,\,c_1,\,c_2,\,c_3,\dots,c_n,\quad\text{and}\quad 1,\,-c_1,\,c_2,-\,c_3,\dots,c_n$$ must have an even number of sign changes.