Roots of a polynomial in $x$ and $e^x$

Question

I have equations of the following form:

$$(p_1(y) + p_2(y) x)^2 + p_3(y) = 0$$

Where $x \in \mathbb{R}$, $y = e^x$ and the $p_i$ are polynomials with real coefficients.

Is there any way to say something about the set of roots to this equation?

Specifically:

• some reasonable bound $B$ such that the zeros are $|x| \le B$
• some bound on the number of roots

I'm trying to write a program to solve the equation numerically (analytical is clearly hopeless), but not sure how to proceed without some info about the roots.

If I knew the roots to an $n$th derivative of the equation, I could work my way "up" and find the roots of the original equation too, but not sure if that helps.

Some related problems I've looked at:

• Real roots for exponential-polynomial equations
• Solving a polynomial with a natural log included with other terms for zero

Any help/ideas would be much appreciated.

Answer 1

Solving for $x$ results in

$$x = \dfrac{-p_1(y)\pm\sqrt{-p_3(y)}}{p_2(y)}=\dfrac{-p_1(\exp(x))\pm\sqrt{-p_3(\exp(x))}}{p_2(\exp(x))}=\Phi(x).$$

You might be able to make some statements about the roots if you are able to check the conditions for the Banach fixed-point theorem.