### Article

Kyungpook Mathematical Journal 2021; 61(3): 455-459

**Published online** September 30, 2021

Copyright © Kyungpook Mathematical Journal.

### Oh's 8-Universality Criterion is Unique

Scott Duke Kominers

Harvard Business School, Department of Economics, and Center of Mathematical Sciences and Applications, Harvard University, Rock Center 219, Soldiers Field, Boston, MA 02163, USA

e-mail : kominers@fas.harvard.edu

**Received**: June 8, 2020; **Revised**: August 30, 2020; **Accepted**: November 23, 2020

### Abstract

We partially characterize criteria for the

**Keywords**: *n*-universal lattice, 8-universal lattice, universality criteria, quadratic form, additively indecomposable.

### 1. Introduction

A degree-two homogeneous polynomial in

A rank-

In 1993, Conway and Schneeberger announced their celebrated

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

The set

Kim, Kim, and Oh [7] have proven that

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 _{8}

### 2. Notation and Terminology

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 _{n}_{8}

For a _{1}_{2}_{1}_{2}

For a sublattice ℓ of

with

### 3. Partial Characterization of n-Criterion Sets

In this section, we prove two results that partially characterize the contents of arbitrary

**Proposition 3.1.** _{n}.

_{n}_{k}

We may therefore write

where _{k,i}

represents all of _{n}_{n}

**Proposition 3.2.** it Let

which of course represents all of

If _{i}_{i}_{i}_{i}_{i}

Thus, we have found a lattice that represents all of

**Remark 3.3.** It is clear that direct analogues of Propositions 3.1 and 3.2 hold in the more general setting of _{n}

### 4. Uniqueness of The 8-Criterion Set

Oh [11] obtained the following

The set

_{8}

Corollary 4.2, when combined with Theorem 4.1, shows that

### Acknowledgements

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.

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