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My question: Is there a known characterization for polynomials $P\in \mathbb{R}[x,y]$ with the property that $$P(x,y) = 0 \wedge P(y,x) =0 \Rightarrow x = \overline{y}$$ and the number of the solutions to the system is finite?

After a change of variables, one have the equivalent condition for $Q\in \mathbb{R}[x,y]$ that $$Q(x,iy) = 0 \wedge Q(x,-iy) =0 \Rightarrow x,y \in \mathbb{R}.$$

This is related to stable polynomials, but not quite. (There is a nice characterization of real stable polynomials, by Lax).

There exist plenty of such polynomials, for example, one could construct a series of such polynomials recursively $$P_n = x P_{n-1} - y P_{n-2}+P_{n-3}$$ with start values $P_0 = 0,P_1 = 0, P_2 = 1.$

This type of polynomials arise naturally from certain determinants of Toeplitz matrices.

EDIT:

Clearly, if $(x,y)$ is a common root,then $(y,x), (\bar{x},\bar{y}), (\bar{y},\bar{x})$ are also roots (by symmetry, and realness of coefficients).

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Are these the same polynomials you asked about at mathoverflow.net/questions/78777/… ? –  Kevin P. Costello Oct 21 '11 at 21:02
    
Yes, I just realized I've asked a related question before. However, I now have a much larger family, given by different recursions (of different lengths). The one in 78777 is just a special case. Seems like I need to refresh my memory about the answer on the question I asked about 6 months ago... –  Per Alexandersson Oct 21 '11 at 21:39

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