BUILT-IN MATHEMATICA SYMBOL

LatticeReduce

gives a reduced basis for the set of vectors

.

The elements of the can be integers, Gaussian integers, or Gaussian rational numbers.

Find the reduced norm basis for a lattice:

Out[1]=

Starting with trivial integer linear relationships, LatticeReduce can produce more interesting ones:

Find integer linear relationships for

and

of the form

:

LatticeReduce preserves linear relationships, and the third row provides

,

, and

:

Find polynomial relationships

for

:

The trivial initial relationships:

The reduced relationships:

The first relationship:

Find linear relationships x₀+x₁ ArcTan[1]+x₂ ArcTan[1/5]+x₃ ArcTan[1/239]==0:

Initial trivial relationships:

Reduced relationships:

The first relationship:

LatticeReduce produces a new reduced basis for the same lattice:

The product of the norms will decrease:

The determinant or volume of the generator cell is preserved:

The lattice is generated by

, but also by

produced by LatticeReduce:

The original cell is pink, and the one produced by LatticeReduce is cyan:

The set of vectors must have rational or Gaussian rational coefficients: