A degree-two homogeneous polynomial in n independent variables is called a quadratic form (or just form) of rank n. For a rank-n quadratic form Q(x1,…,xn)=∑ i,jaijxixj (where aij=aji), the matrix given by L=(aij) is the Gram Matrix of a ℤ-lattice L equipped with a symmetric bilinear form B(⋅,⋅) such that B(L,L)⊆ℤ. Then, Q(x)=xTLx=B(Lx,x) for x∈ℝn.

A rank-n quadratic form Q is said to represent an integer k if there exists an x∈ℤn such that Q(x)=k. More generally, a ℤ-lattice L represents another ℤ-lattice ℓ if there exists a ℤ-linear, bilinear form-preserving injection l→L. A quadratic form is called universal if it represents all positive integers. Analogously, a lattice is called n-universal if it represents all rank-n positive-definite integer-matrix ℤ-lattices. Connecting the two notions of universality, we observe that a rank-n quadratic form Q is universal if and only if it is 1-universal, as for an integer k,

In 1993, Conway and Schneeberger announced their celebrated Fifteen Theorem, giving a criterion characterizing the universal positive-definite integer-matrix quadratic forms. Specifically, they showed that any positive-definite integer-matrix form that represents the set of nine critical numbers

is universal (see [1, 3]). Kim, Kim, and Oh [6] presented an analogous criterion for 2-universality, showing that a positive-definite integer-matrix lattice is 2-universal if and only if it represents the set of forms

Oh [11] gave a similar criterion for 8-universality, which we state in Theorem 4.1 of Section 4.

A set S of rank-n lattices having the property that a lattice L is n-universal if and only if L represents every lattice in S is called an n-criterion set. Thus, for example, the set S2 obtained by Kim, Kim, and Oh [6] is a 2-criterion set and the set of integers found by Conway [3] naturally gives the 1-criterion set

The set S1 is known to be the unique minimal 1-criterion set (see [7]), in the sense that if S1′ is a 1-criterion set, then S1⊆S 1′. The author [9] obtained an analogous uniqueness result for the 2-criterion set S2.

Kim, Kim, and Oh [7] have proven that n-criterion sets exist for all positive integers n. However, the problems of finding and determining the uniqueness of these sets have proven to be difficult (see the discussion in [7]). Here, we advance both problems: We obtain two simple (partial) characterization results for arbitrary n-criterion sets, from which we obtain the uniqueness of Oh's 8-universality criterion as a corollary.

Since we first circulated this paper, there has been renewed attention in characterizing criterion sets: Elkies, Kane, and the author [5] identified several families of lattices for which there exist multiple universality criteria of different sizes, including one based on the ℤn and E₈ lattices that builds on our work here. More recently, Lee [10] and Kim, Lee, and Oh [8] showed that the minimal n-criterion sets are not unique for n≥ 9, and introduced an elegant theory of recoverable lattices that substantially generalizes [5]. (See also recent work of Chan and Oh [2] characterizing classes of exceptional sets for rank-n quadratic forms, which in some sense can be thought of as building blocks for criterion sets.)

We use the lattice-theoretic language of quadratic form theory. A complete introduction to this approach may be found in [12]. In addition, we use the lattice notation of [4], under which I_n is the rank-n lattice of the form 1,…,1 and E₈ is the unique even unimodular lattice of rank 8.

For a ℤ-lattice (or hereafter, just lattice) L with basis {x1,…,xn}, we write L≅ℤx1+⋯+ℤxn. If L is of the form L=L1⊕L2 for sublattices L₁ and L₂ of L with B(L1,L2)=0, then we write L≅L1⊥L2 and say that L₁ and L₂ are orthogonal.

For a sublattice ℓ of L1⊥L2 that can be expressed in the form

with xi,j∈Li, we denote l(Li):=ℤxi,1+⋯+ℤxi,n. We naturally extend this notation to lattices ℓ represented by L1⊥L2. We then say that a lattice is additively indecomposable if either l(L1)≅0 or l(L2)≅0 whenever L1⊥L2 represents ℓ. Otherwise, we say that ℓ is additively decomposable.

In this section, we prove two results that partially characterize the contents of arbitrary n-criterion sets.

Proposition 3.1. Any n-criterion set must include the lattice I_n.

Proof. If T is a finite, nonempty set of rank-n lattices not containing I_n, then every lattice T∈T may be written in the form T≅Ik⊥T′, where 0≤k<n, the sublattice T' is of rank n-k, and the first minimum of T' is larger than 1. Indeed, any I_k-sublattice of T is unimodular and therefore splits T; the condition on T' follows from Minkowski reduction.

We may therefore write T in the form

where 0<|T|=∑ k=0 n−1ik and each T_k,i is a rank-(n-k) lattice with first minimum greater than 1. Then, the lattice

represents all of T but does not represent I_n. It follows that T is not an n-criterion set; hence, any n-criterion set must contain I_n.

Proposition 3.2. it Let E be the set of additively indecomposable unimodular lattices of rank n. If E≠∅, then any n-criterion set must include at least one lattice E∈E.

Proof. Suppose that E≠∅. If T={Ti}i=1k is a finite, nonempty set of rank-n lattices with T∩E=∅, then every lattice Ti∈T is either additively decomposable or not unimodular (or both). Now, we consider the lattice

which of course represents all of T by construction.

If T1⊥⋯⊥Tk were to represent some E∈E, then under any such representation we would have E(Ti)≅0 for all but one i (with 1≤i≤k) because E is additively indecomposable. Then, for some i (again, with 1≤i≤k), the lattice T_i would represent E. In that case, as E is unimodular, the associated sublattice of T_i would split T_i as Ti≅E⊥T′--and since both E and T_i are of rank n, we would have T′≅0; hence, Ti≅E. But this is impossible because T_i is either additively decomposable or not unimodular, whereas E∈E is both additively indecomposable and unimodular.

Thus, we have found a lattice that represents all of T but cannot represent any E∈E. As E≠∅ by hypothesis, we see that T must not be an n-criterion set; the result follows.

Remark 3.3. It is clear that direct analogues of Propositions 3.1 and 3.2 hold in the more general setting of S-universal lattices discussed in [7]. In particular, suppose that S is an infinite set of lattices. Then, if n=maxk:Ik∈S>0, any finite set SS⊂S with the property that a lattice L represents every l∈S if and only if L represents every l∈SS must contain I_n. Similarly, such a set SS must contain an additively indecomposable unimodular lattice if S does.

Oh [11] obtained the following 8-criterion set.

Theorem 4.1.([11, remark on Theorem 3.1]) The set S8={I8,E8} is an 8-criterion set.

The set S8 is clearly a minimal 8-criterion set, as for each l∈S8 there is a lattice that represents S8∖l but does not represent ℓ. (The single lattice in S8∖l suffices.) Meanwhile, our characterization results imply the following corollary, which strengthens Theorem 4.1.

Corollary 4.2. Every 8-criterion set must contain S8 as a subset.

Proof. As E₈ is the unique additively indecomposable unimodular lattice of rank 8, the result follows directly from Propositions 3.1 and 3.2.

Corollary 4.2, when combined with Theorem 4.1, shows that S8 is the unique minimal 8-criterion set.

The author is grateful to Pablo Azar, Noam D. Elkies, Andrea J. Hawksley, Sonia Jaffe, Paul M. Kominers, and especially Ravi Jagadeesan for helpful comments and suggestions, and particularly thanks an anonymous referee for pointing out a problem with the original form of Proposition 3.2.