I believe that there is no common theory for finding roots of polynomial sum. In my case I have $$P_{n}(x)+AQ_{n}(x)$$. I am wondering how roots of this sum depend on $A$?
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$\begingroup$ What is assumed about $P_n$, $Q_n$ and their roots? $\endgroup$– Andrey RekaloCommented Jun 30, 2010 at 16:43
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4$\begingroup$ Highly non-trivial techniques to do this were developed by Goldstein and Schlag in math.uchicago.edu/~schlag/papers/gaps.pdf and math.uchicago.edu/~schlag/papers/equi.pdf . They had quite a specific problem in mind, so this might not be too helpful for you. The buzzword there is "Jensen formula". $\endgroup$– HelgeCommented Jun 30, 2010 at 19:43
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$\begingroup$ in my case $P_n$ and $Q_n$ are known rekursiv polynomials of degree $n$.Positive.A>0. So all roots are complex $\endgroup$– vilvarinCommented Jul 1, 2010 at 17:28
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$\begingroup$ would $P_n(x) + AxP_n(x)$ makes the problem simpler ? $\endgroup$– yupbankCommented Nov 29, 2021 at 17:58
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2 Answers
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If they are complex polynomials or can be treated as such, then you could apply Rouche's theorem, where the location of the zeros is determined by the dominant polynomial within the sum. ("Walk the dog on the leash")
Possibly related: you could use the Wronskian to determine the values of A that make $P_n(x)$ and $Q_n(x)$ linearly independent.
Your question is related to Mason's theorem. There are a few papers which explore this specifically
- MR1923392 (2003j:30012) Kim, Seon-Hong . Factorization of sums of polynomials. Acta Appl. Math. 73 (2002), no. 3, 275--284.
- MR2103113 (2005h:30011) Kim, Seon-Hong . On zeros of certain sums of polynomials. Bull. Korean Math. Soc. 41 (2004), no. 4, 641--646
- MR1911767 (2003d:11036) Pintér, Á. Zeros of the sum of polynomials. J. Math. Anal. Appl. 270 (2002), no. 1, 303--305.
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Though in general you won't have a closed-form expression for the roots of your polynomials, it's possible to write down perturbation series for roots of a polynomial in a single variable in terms of the coefficients. These are basically the Puiseux series mentioned in this question.
This paper by Bernd Sturmfels (MR) sketches out the "global picture" of such series, though it's fairly complicated and I personally am not clear on whether there's a simple algorithm to decide which is the proper choice of series that will converge. See also the article "Algebraic equations and hypergeometric series" by M Passare, A Tsikh in the book: The Legacy of Niels Henrik Abel (MR).
What I've just written is probably a little unclear so I'll describe the simplest example. Suppose you'd like to write down a series for the roots of $a_2x^2+a_1x+a_0=0$. There are a pair of series which converges when $\left|\frac{a_1^2}{4a_0a_2}\right|<1$ and a pair which converges when $\left|\frac{a_1^2}{4a_0a_2}\right|>1$, and you can derive the first pair of series by treating $a_1x$ as a perturbation to the equation $a_2x^2+a_0=0$ and you can derive one of the second pair of series by treating $a_2x^2$ as a perturbation to $a_1x+a_0=0$ and the other by treating $a_0$ as a perturbation to the equation $a_2x^2+a_1x=0$.
By plugging in the coefficients of $P_n(x)+AQ_n(x)$ into the appropriate series I just described and looking at the leading order terms as functions of $A$, you will be able to derive the scaling of the corrections to the roots of $P_n(x)$.
Apologies for the rather unexplicit answer, but this is just at the limit of what I understand.
