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- - **Small problem in six-dimensional space**
(*https://www.mersenneforum.org/showthread.php?t=10648*)

Small problem in six-dimensional spaceI'm considering the codimension-1-in-P5-object defined by
x^6+y^6+z^6 = u^6+v^6+w^6 with the obvious solutions-by-permutation removed and GCD(x,y,z,u,v,w)=1. I've searched for z,w<2000, and am finding that the number of these points with z,w<N seems to be slightly more than linear in N, so I suspect there are some families of points lurking. Suspiciously many of the points I've found have x+y+u=z+v+w, for example. [url]http://www.jstor.org/pss/2005335[/url] gives a set of homogeneous quartics parameterising some solutions, but clearly not all solutions since the quartics happen also to satisfy x^2+y^2+z^2=u^2+v^2+w^2. What are the right sort of questions to ask about the set of points on a high-dimensional variety? |

[QUOTE=fivemack;142149]I'm considering the codimension-1-in-P5-object defined by
x^6+y^6+z^6 = u^6+v^6+w^6 <snip> What are the right sort of questions to ask about the set of points on a high-dimensional variety?[/QUOTE] Is there an associated L-function? Is there a motive? What does the j-invariant look like? Is the variety even Abelian? How many localizations yield singularities? (i.e. how many primes have bad reduction) |

[QUOTE=R.D. Silverman;142240]Is there an associated L-function? Is there a motive?
What does the j-invariant look like? Is the variety even Abelian? How many localizations yield singularities? (i.e. how many primes have bad reduction)[/QUOTE] And I stupidly forgot perhaps the most obvious question: What is its genus? i.e. if viewed as a Riemann surface over C, how many holes does the surface have? Then one might study conformal maps of that surface. |

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