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| In [[mathematics]], a '''diagonal form''' is an algebraic form ([[homogeneous polynomial]]) without cross-terms involving different [[indeterminate (variable)|indeterminates]]. That is, it is
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| :<math>\Sigma a_i {x_i}^m\ </math>
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| for some given degree ''m'', summed for 1 ≤ ''i'' ≤ ''n''.
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| Such forms ''F'', and the [[hypersurface]]s ''F'' = 0 they define in [[projective space]], are very special in geometric terms, with many symmetries. They also include famous cases like the [[Fermat curve]]s, and other examples well known in the theory of [[Diophantine equation]]s.
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| A great deal has been worked out about their theory: [[algebraic geometry]], [[local zeta-function]]s via [[Jacobi sum]]s, [[Hardy-Littlewood circle method]].
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| ==Examples==
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| :<math>X^2+Y^2-Z^2 = 0</math> is the [[unit circle]] in ''P''<sup>2</sup>
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| :<math>X^2-Y^2-Z^2 = 0</math> is the [[unit hyperbola]] ''P''<sup>2</sup>. | |
| :<math>x_0^3+x_1^3+x_2^3+x_3^3=0</math> gives the Fermat [[cubic surface]] in ''P''<sup>3</sup> with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (''x'' : ''ax'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates. | |
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| :<math>x_0^4+x_1^4+x_2^4+x_3^4=0</math> gives a [[K3 surface]] in ''P''<sup>3</sup>.
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| {{DEFAULTSORT:Diagonal Form}}
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| [[Category:Homogeneous polynomials]]
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| [[Category:Algebraic varieties]]
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Revision as of 08:08, 17 February 2014
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