Ternary Classical Universal Quadratic Form

• Ternary classical universal quadratic forms are three-variable positive definite integral forms that aim to represent every nonnegative integer under strict local conditions.
• Local techniques, including spinor genus analysis and Jordan splittings, precisely identify when these forms fail to achieve complete universality.
• Recent algorithmic and computational advances enable effective testing of almost universality and reveal key invariants influencing quadratic form representations.

A ternary classical universal quadratic form is a positive definite integral quadratic form in three variables (rank 3) that represents every nonnegative integer. Classical universal forms are a central object in the arithmetic theory of quadratic forms, especially emphasizing universality (unconditional representation of all relevant numbers) over the integers and, more generally, over rings of integers in number fields. For ternary quadratic forms (i.e., forms in three variables), the landscape is sharply constrained by both local representation principles and deep global obstructions, as articulated in the theory initiated by Gauss, Dickson, and developed through modern local-global and spinor genus techniques.

1. Definitions and Historical Context

A quadratic form $Q$ in three variables over $\mathbb{Z}$ or over the ring of integers $\mathcal{O}_K$ of a totally real number field $K$ is classical if all off-diagonal coefficients are even (when working over number fields: $\alpha_{ij}\in 2\mathcal{O}_K$ for $i\neq j$). Universality requires that for every nonnegative integer $n$ (or totally positive integer $\alpha\in\mathcal{O}_K^+$), $n$ is represented by $Q$, i.e., there exists $(x_1,x_2,x_3)\in\mathbb{Z}^3$ (or $\mathcal{O}_K^3$) with $Q(x_1,x_2,x_3)=n$ (or $= \alpha$).

The theory is underpinned by several classical theorems:

• Lagrange’s Four Squares Theorem: every $n\in\mathbb{N}$ is represented by $x^2 + y^2 + z^2 + w^2$.
• Gauss’s Three Triangular Numbers Theorem: every $n\in\mathbb{N}$ is a sum of three triangular numbers, which can be written as an inhomogeneous ternary quadratic form.
• Legendre’s Three Squares Theorem: $n$ is represented by $x^2 + y^2 + z^2$ if and only if $n \not\equiv 7\pmod{8}$.

However, the Local Square Theorem and subsequent work revealed that no ternary quadratic form is universal: every such form must miss infinitely many positive integers.

2. Local-Global Principles and Conductors

For inhomogeneous ternary quadratic polynomials $f(x) = Q(x) + 2B(v,x) + c$, the conductor $m$ is defined as the minimal positive integer for which $m \cdot v \in N$ (where $N$ is the lattice corresponding to $Q$, and $B$ is the associated bilinear form). The “universal” or “almost universal” behavior is controlled by both the global form and the local representation conditions at each prime.

A precise “if and only if” characterization for almost universal ternary quadratic polynomials with odd prime power conductor is developed in (Haensch, 2014). The main result (Theorem 7) asserts that universality (up to finitely many small exceptions) holds exactly when:

• For every prime $q\neq p$, the local lattice $N_q$ represents all $q$-adic integers.
• For $q=p$, the lattice $M_q$ associated to the coset $v+N$ is locally universal.

If $p\equiv7\pmod{8}$, additional arithmetic conditions involving the $p$-adic valuation of the discriminant $d_N$, the square-free part $sf(d_N)$, and the behavior of the spinor norm group at $p$ are required to eliminate primitive spinor exceptions.

3. Local, Spinor, and Genus Theory

Local obstructions are paramount: for a ternary quadratic form to be universal or almost universal, every local component must represent all relevant classes. The theory of spinor genus refines genus theory, capturing the subtle distinction between local representability and global failure due to spinor exceptions.

The main obstruction comes from the possibility that a number is represented by all local genera, but not globally, because of spinor genus constraints. This is especially delicate at $p=2$ and when the quadratic form is highly anisotropic at some local place. The conditions (a)–(d) in Theorem 7 of (Haensch, 2014) express precisely when spinor exceptions can or cannot occur.

4. Arithmetic Progressions and Coprime Universality

Ternary quadratic forms may be “universal” on arithmetic progressions or under coprimality constraints. $(d,r)$-universality and $(k,\ell)$-universal forms, as defined in (Wu et al., 2018) and (Hejda et al., 2019), respectively, are diagonal ternary forms representing all $d n+r$ ($n\geq 0$) or all $k n+\ell$ ($k\nmid\ell$) numbers.

Almost $(k, \ell)$-universality is shown to always occur for diagonal ternary forms when $k \nmid \ell$ (Hejda et al., 2019), but true universality is rare and may occur for only finitely many primes (Hejda et al., 2019), with extensive computer evidence supporting this.

Similarly, the characterization of coprime-universal forms (representing all $n$ coprime to a prime $p$) generalizes the 290-Theorem, with explicit analytic and arithmetic criteria for existence (Bordignon et al., 3 Jun 2024). For certain small $p$, possible ternary (three-variable) coprime-universal forms exist, but for sufficiently large $p$, universality demands at least four variables.

5. Number Field and Local Field Generalizations

The extension of ternary classical universal form theory to totally real fields is covered in depth in (Krásenský et al., 2019, Kala et al., 2021), and (Kala et al., 22 Oct 2025). These works establish that, except for a handful of fields (notably $\mathbb{Q}(\sqrt{5})$ when the discriminant is odd), no universal ternary classical quadratic form exists over the integer ring of a totally real number field. The obstruction is linked to additively indecomposable elements and the arithmetic of totally positive units.

The local theory (over dyadic local fields), articulated in (Beli, 2020, He et al., 2022), and (He, 2022), provides necessary and sufficient invariants (Jordan splittings, BONGs, and norm/scale conditions) for $3$-universality. Over unramified dyadic fields, classical ternary universal forms are characterized via modularity, unimodularity, and discriminant conditions of Jordan components.

6. Algorithmic, Computational, and Representational Consequences

Explicit characterization results (e.g., Theorem 7 (Haensch, 2014), Siegel formula applications (Berkovich et al., 2014), genus and spinor genus theory) enable algorithmic verification of universality or almost universality for a given ternary quadratic form. The computation of local densities and analysis of exceptional classes (e.g., residue classes mod 8 or 16 (Jafari et al., 2021)) concretely identifies nonrepresentable numbers and explains the necessity of four variables for true universality over $\mathbb{Z}$.

Ternary forms remain a powerful model for understanding local-global phenomena, the arithmetic of number fields, and connections to class numbers (via Hurwitz class numbers, theta series coefficients, and Eisenstein series representations (Luo et al., 27 Feb 2024)).

7. Summary Table of Existence/Obstruction

Context	Universal Ternary Form?	Key Invariant/Obstruction
$\mathbb{Z}$	No (infinitely many exceptions)	Residue classes mod $8$, $16$ (Jafari et al., 2021)
Arithmetic Progression	Yes, for some $(d,r)$	Local densities + genus theory (Wu et al., 2018)
$(k,\ell)$, $k\nmid \ell$	Almost universal form exists	Anisotropy, spinor exception analysis (Hejda et al., 2019)
$\mathcal{O}_K$, totally real	Rare, only for select $K$	Additive indecomposables, units (Kala et al., 22 Oct 2025)
Dyadic local field $F$	Explicitly classified	Jordan splitting/BONG invariants (He et al., 2022)

8. Implications and Research Directions

The framework for ternary classical universal quadratic forms provides both deep theoretical insights and concrete computational tools. The effective characterization in (Haensch, 2014) allows practical testing, while generalizations into number fields and local fields illuminate the role of unit groups, arcs of arithmetic progression, and modular invariants. The theory is crucial for understanding the limitations of universality, the structure of representation sets, and the transcendence from homogenous forms to inhomogeneous polynomials and lattice coset representations. Ultimately, results confirm that true universality (in three variables) is rare or impossible except for very select arithmetic settings, and that almost universal forms exhibit a rich, locally constrained, globally subtle behavior.