# A Lecture by Polya on Symmetric Algebraic Equations with an Unexpected Conclusion

Sometime in 1980 George Polya gave a lecture at the University of Minnesota about solutions of algebraic equations that have symmetry in the appearance of the variables in the equation (any permutation of the variables resulted in the same system).

My recall is far from complete, but I suspect he began with a 2-dimensional system like x + y = 1 and $x^2 + y^2 = 4$. Simple algebra without computational error shows this system has solutions (x,y) = $([1+\sqrt{7}]/2) , [1-\sqrt{7}]/2)$ and (x,y) = $([1-\sqrt{7}]/2) , [1+\sqrt{7}]/2)$. The solution sets for x and for y are equal.

He generalized to a class of 2-dimensional systems whose solution sets for each variable were the same set. I recall thinking his conclusions were obvious (my arrogance and/or my ignorance).

He repeated in three dimensions, and then in four, while I wondered why I had come to the lecture.

But Polya, in his eighties at the time, was still a master. Eventually he displayed a system (in six or seven dimensions) with the required symmetry, in which the solution sets for some of the variables were not equal.

I assume he gave this lecture elsewhere. Does anyone know more details?

• I once heard a lecture by Leo Moser with some of these features. The (erroneous) principle that symmetry in a problem implies symmetry in the solution he called "The Principle of Insufficient Reason". Maybe with that name you can try finding what Polya wrote on it. – Gerald Edgar Nov 6 '15 at 14:28
• Something seems to be missing, in your example the solutions are $(1\pm\sqrt{7})/2$ in both orders and $x\neq y$. It follows from symmetry that any permutation of a solution is another solution but it is rather rare that all values are necessarily equal. – Conifold Nov 6 '15 at 17:03
• Thank you for each of the comments. To Connifold. I agree that what I wrote was not well-stated. I will try to edit my original question so it will make sense to future readers. – euler1944 Nov 7 '15 at 17:07
• A paper on this idea, but not by Polya: maa.org/sites/default/files/images/images/upload_library/22/…. – KCd Mar 17 '17 at 18:28
• A similar non-symmetry occurs more familiarly with, for example, constant-coefficient linear differential equations, where the coefficients are translation-invariant functions, but the solutions are (mostly) not. (This segues into Floquet Theory, also.) – paul garrett Jun 18 '19 at 19:39