2
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.