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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:

Any help/ideas would be much appreciated.

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  • $\begingroup$ what is the degree of $p_3$? $\endgroup$ – uniquesolution Feb 22 at 21:50
  • $\begingroup$ It might be worthy to check the derivative - when is it eventually positive/negative. Note that $$ \frac {\text d}{\text dx} p(y) = yp'(y) $$ which doesn't have the free term. However, it's harder to solve the derivative for $x$, as you lose the square. $\endgroup$ – eudes Feb 23 at 0:19
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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.

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  • $\begingroup$ How about putting $\log y$ instead of $x$? And asymptotically $$ (-p_1 \pm \sqrt p_3)/p_2 \approx \text{const} \cdot y ^\wedge [\max(\deg p_1, \ 1/2 \ \deg p_3) - \deg p_2], $$ for large $y$ unless $\deg p_1 = \: 1/2 \ \deg p_3$ and leading coefficients of $p_1$ and $p_3$ are appropriate. For small $y$, consider $\log (1/y)$ and polynomials in $1/y$ (degrees will possibly change). $\endgroup$ – eudes Feb 22 at 23:55

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