Editing Kissing number (section)

==Mathematical statement==
The kissing number problem can be stated as the existence of a solution to a set of [[Inequality (mathematics)|inequalities]]. Let <math>x_n</math> be a set of ''N'' ''D''-dimensional position vectors of the centres of the spheres. The condition that this set of spheres can lie round the centre sphere without overlapping is:<ref>Numbers ''m'' and ''n'' run from 1 to ''N''. <math>x = (x_n)_N</math> is the sequence of the ''N'' positional vectors. As the condition behind the second universal quantifier (<math>\forall</math>) does not change if ''m'' and ''n'' are exchanged, it is sufficient to let this quantor extend just over <math>m,n:m<n</math>. For simplification the sphere radiuses are assumed to be 1/2.</ref>
<math display=block>\exist x\ \left\{ \forall_n \{x_n^\textsf{T} x_n = 1\} \land \forall_{m,n: m \neq n} \{ (x_n - x_m)^\textsf{T}(x_n - x_m) \geq 1\} \right\}</math>

Thus the problem for each dimension can be expressed in the [[existential theory of the reals]]. However, general methods of solving problems in this form take at least [[exponential time]] which is why this problem has only been solved up to four dimensions. By adding additional variables, <math>y_{nm}</math> this can be converted to a single [[quartic function|quartic equation]] in ''N''(''N''&nbsp;−&nbsp;1)/2 + ''DN'' variables:<ref>Concerning the matrix <math>y = (y_{mn})_{N\times{N}}</math> only the entries having ''m''&nbsp;<&nbsp;''n'' are needed. Or, equivalent, the matrix can be assumed to be antisymmetric. Anyway the matrix has just''N''(''N''&nbsp;−&nbsp;1)/2 free scalar variables. In addition, there are ''N'' ''D''-dimensional vectors ''x''<sub>''n''</sub>, which correspondent to a matrix <math>x = (x_{nd})_{N \times D}</math> of ''N'' column vectors.</ref>
<math display=block> \exist xy\ \left\{ \sum_n \left(x_n^\textsf{T} x_n - 1\right)^2 + \sum_{m,n: m<n}\Big( (x_n - x_m)^\textsf{T} (x_n - x_m) - 1 - (y_{nm})^2 \Big)^2 = 0 \right\} </math>

Therefore, to solve the case in ''D''&nbsp;=&nbsp;5 dimensions and ''N''&nbsp;=&nbsp;[[D5 lattice|40]]&nbsp;+&nbsp;1 vectors would be equivalent to determining the existence of real solutions to a quartic polynomial in 1025 variables. For the ''D'' = 24 dimensions and ''N''&nbsp;=&nbsp;[[Leech lattice|196560]]&nbsp;+&nbsp;1, the quartic would have 19,322,732,544 variables. An alternative statement in terms of [[distance geometry]] is given by the distances squared <math>R_{mn}</math> between the ''m''<sup>th</sup> and ''n''<sup>th</sup> sphere:
<math display=block> \exist R\ \{ \forall_n \{R_{0n} = 1 \} \land \forall_{m,n: m<n} \{ R_{mn} \geq 1\} \}</math>

This must be supplemented with the condition that the [[Cayley–Menger determinant]] is zero for any set of points which forms a (''D''&nbsp;+&nbsp;1) simplex in ''D'' dimensions, since that volume must be zero. Setting <math>R_{mn} = 1 + {y_{mn}}^2</math> gives a set of simultaneous polynomial equations in just ''y'' which must be solved for real values only. The two methods, being entirely equivalent, have various different uses. For example, in the second case one can randomly alter the values of the ''y'' by small amounts to try to minimise the polynomial in terms of the&nbsp;''y''.