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 : firstname.lastname@example.org
Received: June 8, 2020; Revised: August 30, 2020; Accepted: November 23, 2020
We partially characterize criteria for the
Keywords: n-universal lattice, 8-universal lattice, universality criteria, quadratic form, additively indecomposable.
A degree-two homogeneous polynomial in
In 1993, Conway and Schneeberger announced their celebrated
is universal (see [1, 3]). Kim, Kim, and Oh  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  gave a similar criterion for 8-universality, which we state in Theorem 4.1 of Section 4.
Kim, Kim, and Oh  have proven that
Since we first circulated this paper, there has been renewed attention in characterizing criterion sets: Elkies, Kane, and the author  identified several families of lattices for which there exist multiple universality criteria of different sizes, including one based on the
2. Notation and Terminology
We use the lattice-theoretic language of quadratic form theory. A complete introduction to this approach may be found in . In addition, we use the lattice notation of , under which
For a sublattice ℓ of
3. Partial Characterization of n-Criterion Sets
In this section, we prove two results that partially characterize the contents of arbitrary
We may therefore write
represents all of
Proposition 3.2. it Let
which of course represents all of
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
4. Uniqueness of The 8-Criterion Set
Oh  obtained the following
Corollary 4.2, when combined with Theorem 4.1, shows that
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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