The graph of solutions to $x^4+y^4=x^2+y^2$ is as follows:
Clearly the origin is also part of the graph. Is there a name for these kinds of points?
2026-04-13 12:33:58.1776083638
Graph of $x^4+y^4=x^2+y^2$
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Your point is an isolated point of the real locus of the quartic algebraic curve $X^4+Y^4-X^2-Y^2=0$. But when you’re doing algebraic geometry, it’s always valuable to have the complex reality in mind. If you make the simple coordinate transformation $x=X, y=iY$, then you get $x^4+y^4-x^2+y^2=0$, whose real locus is a familiar-looking butterfly curve. The origin is still singular, but not isolated. You expect that if the defining polynomial is irreducible, then the (complex) locus will be connected.