: Mathematics: Bring's curve

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geometry > complex geometry > complex manifold > Riemann surface > Bring's curve

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Bring's curve

In mathematics, Bring's curve (also called Bring's surface and, by analogy with the Klein quartic, the Bring sextic) is the curve in ${\\displaystyle \\mathbb {P} ^{4}}$ cut out by the homogeneous equations
${\\displaystyle v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.}$
It was named by Klein (2003, p.157) after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund. Note that the roots x_i of the Bring quintic ${\\displaystyle x^{5}+ax+b=0}$ satisfies Bring's curve since ${\\displaystyle \\sum _{i=1}^{5}x_{i}^{k}=0}$ for ${\\displaystyle k=1,2,3.}$

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