Geometry » Algebraic Geometry

Jacobian Conjecture ★★★

Author(s): Keller

Conjecture   Let $ k $ be a field of characteristic zero. A collection $ f_1,\ldots,f_n $ of polynomials in variables $ x_1,\ldots,x_n $ defines an automorphism of $ k^n $ if and only if the Jacobian matrix is a nonzero constant.

Keywords: Affine Geometry; Automorphisms; Polynomials

Posted by Charles
updated March 4th, 2021
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Graph Theory » Coloring » Vertex coloring

Counting 3-colorings of the hex lattice ★★

Author(s): Thomassen

Problem   Find $ \lim_{n \rightarrow \infty} (\chi( H_n , 3)) ^{ 1 / |V(H_n)| } $.

Keywords: coloring; Lieb's Ice Constant; tiling; torus

Posted by mdevos
updated July 5th, 2008
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Geometry

Dense rational distance sets in the plane ★★★

Author(s): Ulam

Problem   Does there exist a dense set $ S \subseteq {\mathbb R}^2 $ so that all pairwise distances between points in $ S $ are rational?

Keywords: integral distance; rational distance

Posted by mdevos
updated July 4th, 2008
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