A383643 Number of n-dimensional additively indecomposable positive definite integral lattices (or quadratic forms).

1, 0, 0, 0, 0, 1, 1, 1, 2

Offset: 1

Author

Robin Visser, May 09 2025

Keywords

• nonn
• hard
• more

Comments

A positive definite integral lattice (or quadratic form) is additively indecomposable if it cannot be written as a sum of two nonzero positive semidefinite integral lattices (or quadratic forms). Any additively indecomposable lattice is also (orthogonally) indecomposable, although the converse need not hold.

By computing all additively indecomposable lattices of determinant up to 100, Opgenorth gave the lower bounds a(10) >= 7 and a(11) >= 13. Eisenbarth showed the lower bounds a(12) >= 29, a(13) >= 9, a(14) >= 10, a(15) >= 9, and a(16) >= 5.

Examples

			For n <= 8, the only n-dimensional additively indecomposable positive definite lattices are Z (of dimension 1), E6 (of dimension 6), E7 (of dimension 7), and E8 (of dimension 8).
For n = 9, the a(9) = 2 additively indecomposable rank 9 positive definite lattices were computed by Opgenorth. These are the two lattices with Gram matrices:
  [ 2  1  1  1  1  1  1  1  4]   [ 2  1  1  1  2  2  2  2  6]
  [ 1  2  1  1  1  1  1  1  4]   [ 1  2  1  1  2  2  2  2  6]
  [ 1  1  2  1  1  1  1  1  4]   [ 1  1  2  1  2  2  2  2  6]
  [ 1  1  1  2  1  1  1  1  4]   [ 1  1  1  2  2  2  2  2  6]
  [ 1  1  1  1  2  1  1  1  4]   [ 2  2  2  2  5  4  4  4 12]
  [ 1  1  1  1  1  2  1  1  4]   [ 2  2  2  2  4  5  4  4 12]
  [ 1  1  1  1  1  1  2  1  4]   [ 2  2  2  2  4  4  5  4 12]
  [ 1  1  1  1  1  1  1  2  4]   [ 2  2  2  2  4  4  4  5 12]
  [ 4  4  4  4  4  4  4  4 15],  [ 6  6  6  6 12 12 12 12 35],
  having determinant 7 and determinant 15 respectively.
		

References

• Jürgen Opgenorth, Additiv unzerlegbare ganzzahlige quadratische Formen in den Dimensionen 9, 10 und 11. Diplomarbeit, RWTH Aachen, 1992.

Links

• Simon Eisenbarth, Additive Zerlegungen von Gittern, Bachelorarbeit, RWTH Aachen, 2014.
• Paul Erdős and Chao Ko, On definite quadratic forms, which are not the sum of two definite or semi-definite forms, Acta Arith. 3, 102-122 (1938).
• Louis J. Mordell, The representation of a definite quadratic form as a sum of two others, Ann. of Math. (2) 38 (1937), no. 4, 751-757.
• Wilhelm Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory, 47 (1994), no. 3, 273-283.
• Ruiqing Wang, Additively indecomposable positive definite integral lattices, Acta Math. Sin. (Engl. Ser.) 41 (2025), no. 3, 908-924.

Crossrefs

Cf. A380746.

Formula

a(n) >= A380746(n).