Notes - Quaternary Universal Quadratic Forms

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Quaternary Universal Quadratic Forms

Yoshio Mimura

Kobe Pharmaceutical University

1. Introduction. Consider an n-ary positive definite integral quadratic form

Q(x₁,..,x_n) = Sum a_ijx_ix_j (1 <= i <= j <= n) where a_ij's are rational integers. Let V(Q) be the value set of Q, that is,
V(Q) = { Q(x) | x runs over n-dimensional non-zero integral vectors }. By e(Q) we denote the smallest positive integer which cannot be represented by Q. If Q can represent all positive integers, that is, if V(Q) = N, then we say that Q is universal. It is known that Q cannot be universal when n < 4. On the other hand, it is well known that the quaternary unit form x₁²+x₂²+x₃²+x₄² is universal. There are 54 quaternary universal diagonal forms (Ramanujan, Pall) and there are 204 quaternary universal "integral" (that is, all a_ij are even if i is not equal to j) forms (Willerding, Conway, Bhargava). In this note we shall find all quaternary integral forms which represent all integers less than 10000. Hence our List contains all quaternary universal forms. Remark that H. Iwabuchi has proved the universality of the last form of the largest discriminant 4292 in our List.

We consider the n x n symmetric matrix A the (i,j)-element of which is a_ij (but the diagonal elements are 2a_ii's), and define the discriminant disc(Q) of Q to be the determinant of the matrix A. We often identify a form Q and the numbers a_ij's as follows:
[ a₁₁, a₂₂ : a₁₂ ] when n = 2
[ a₁₁, a₂₂, a₃₃ : a₁₂, a₁₃, a₂₃ ] when n = 3
[ a₁₁, a₂₂, a₃₃, a₄₄ : a₁₂, a₁₃, a₁₄, a₂₃, a₂₄, a₃₄ ] when n = 4

2. Binary Subforms
Suppose that Q represents 1 and 2. Then Q represents a binary form x₁² + 2x₂² +ax₁x₂ for some integer a. Since Q is positive definite, we see that a = 0, -1, 1, 2, -2. By a suitable change of variables we can get the following three forms.

Lemma 1. If Q represents 1 and 2, then Q represents one of the following binary forms:
D = 4, [ 1, 1 : 0 ], e = 3
D = 7, [ 1, 2 : 1 ], e = 4
D = 8, [ 1, 2 : 0 ], e = 5

Remark. Let Q be a binary positive definite integral quadratic form which represents 1,2,..,m. Then m <= 4. And Q(x₁,x₂) = x₁² + 2x₂² represents 1,2,3,4.

3. Ternary Subforms
Suppose that Q represents 1,..,5. Then, by Lemma 1, we see that Q represents one of the following ternary forms:

[ 1, 1, 3 : 0, a, b ]
[ 1, 2, 3 : 1, a, b ]
[ 1, 2, 5 : 0, a, b ]

The integers a and b are restricted by the positive definiteness. In the first case the discriminant of the ternary form is 24 - 2(a² + b²) > 0. Similarly we see that 42 -4a² + 2ab - 2b² > 0 in the second case and 80 - 4a² - 2b² > 0 in the third case.

Lemma 2. If Q represents 1,2,3,4,5, then Q represents one of the following 27 ternary forms:

disc a₁₁ a₂₂ a₃₃ a₁₂ a₁₃ a₂₃ e

20 1 1 3 0 1 1 6

22 1 1 3 0 1 0 22

24 1 1 3 0 0 0 6

24 1 2 2 1 1 1 6

26 1 2 2 1 0 1 13

28 1 2 2 1 0 0 5

30 1 2 2 0 0 1 10

32 1 2 2 0 0 0 7

34 1 2 3 1 1 2 17

36 1 2 3 0 1 2 14

38 1 2 3 1 1 1 10

40 1 2 3 1 0 1 5

42 1 2 3 0 1 1 7

42 1 2 3 1 0 0 6

44 1 2 3 0 1 0 10

46 1 2 3 0 0 1 23

48 1 2 3 0 0 0 10

56 1 2 4 0 0 2 7

58 1 2 4 0 1 1 29

62 1 2 4 0 0 1 31

64 1 2 4 0 0 0 14

68 1 2 5 0 1 2 10

72 1 2 5 0 0 2 7

74 1 2 5 0 1 1 29

76 1 2 5 0 1 0 10

78 1 2 5 0 0 1 13

80 1 2 5 0 0 0 10

Proposition 1. Let Q be a ternary positive definite integral quadratic form which represents 1,2,..,m. Then m <= 30. And Q(x₁,x₂,x₃) = x₁² + 2x₂² + 4x₃² + x₂x₃ represents 1,2,..,30.

4. Quaternary forms

If Q is a quaternary positive definite integral quadratic form, then Q has its (reduced) symmetric matrix A:

2a₁₁	a₁₂	a₁₃	a₁₄
a₁₂	2a₂₂	a₂₃	a₂₄
a₁₃	a₂₃	2a₃₃	a₃₄
a₁₄	a₂₂	a₂₃	2a₄₄

where
1 <= a₁₁ <= a₂₂ <= a₃₃ <= a₄₄,
-a₁₁ < a₁₄ <= a₁₁, -a₂₂ < a₂₄ <= a₂₂, 0 <= a₃₄ < a₃₃
0 <= a₂₄ if a₃₄ = 0,
0 <= a₁₄ if a₂₄ = a₃₄ = 0,
and the ternary lattice [ a₁₁, a₂₂, a₃₃ : a₁₂, a₁₃, a₂₃ ] is reduced.
Assume that Q represents 1,..,31. Then, by Lemma 2,
a₁₁=1, a₂₂ <= 2, a₃₃ <= 5, a₄₄ <= 31. We construct all positive definite integral forms for possible a_ij's and delete the forms with e(Q) < 50. Then we have 9648 forms. Then we choose a class representative in each class (6440 classes). Finally we test whether each Q represents 1,2,..,10000. There are four forms which cannot pass the test:

[ 1, 2, 3, 8 : 0, 0, 1, 0, 1, 3 ], D = 660, e = 58,
[ 1, 1, 3, 22 : 0, 0, 1, 0, 0, 0 ], D = 968, e = 110,
[ 1, 2, 4, 29 : 0, 1, 0, 1, 0, 0 ], D = 3364, e = 145,
[ 1, 2, 4, 28 : 0, 0, 0, 1, 0, 1 ], D = 3464, e = 93.

Theorem. If Q is a quaternary positive definite integral quadratic form which represents 1,2,..,145, then Q represents 1,2,..,10000 and it is one of the 6436 forms in List (265KB).

Last updated: 9 March 2001

disc	a₁₁	a₂₂	a₃₃	a₁₂	a₁₃	a₂₃	e
20	1	1	3	0	1	1	6
22	1	1	3	0	1	0	22
24	1	1	3	0	0	0	6
24	1	2	2	1	1	1	6
26	1	2	2	1	0	1	13
28	1	2	2	1	0	0	5
30	1	2	2	0	0	1	10
32	1	2	2	0	0	0	7
34	1	2	3	1	1	2	17
36	1	2	3	0	1	2	14
38	1	2	3	1	1	1	10
40	1	2	3	1	0	1	5
42	1	2	3	0	1	1	7
42	1	2	3	1	0	0	6
44	1	2	3	0	1	0	10
46	1	2	3	0	0	1	23
48	1	2	3	0	0	0	10
56	1	2	4	0	0	2	7
58	1	2	4	0	1	1	29
62	1	2	4	0	0	1	31
64	1	2	4	0	0	0	14
68	1	2	5	0	1	2	10
72	1	2	5	0	0	2	7
74	1	2	5	0	1	1	29
76	1	2	5	0	1	0	10
78	1	2	5	0	0	1	13
80	1	2	5	0	0	0	10