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.