Canonical Degree Six Covariant

• The canonical degree six covariant is a sextic form constructed from a binary quartic and its Hessian, revealing inherent algebraic symmetries.
• It establishes a profound link between binary quartics and ternary quadratic forms via a cubic covariant, facilitating matrix diagonalization.
• The covariant plays an essential role in arithmetic invariant theory, offering criteria for simultaneous diagonalizability and aiding in the parametrization of quartic rings.

A canonical degree six covariant arises in multiple domains of mathematics and mathematical physics, with each instance presenting a distinct role for such covariant structures. In classical invariant theory, it specifically denotes a naturally constructed sextic covariant of a binary quartic form, capturing intricate relationships within the algebraic structure of forms and linking the theory of binary forms to the geometry of ternary quadratic forms (Xiao, 5 Aug 2025). In various covariant formulations of field theory or gravity, the phrase "degree six covariant" may refer to canonical variables associated with six independent degrees of freedom—such as those comprising the independent components of a symmetric rank-two tensor in four dimensions—or to a six-component canonical field for a spin-1 particle multiplet (Simulik, 2014, Kluson et al., 2022). This article concentrates on the mathematical essence, explicit construction, correspondences, and significance of the canonical degree six covariant as detailed in the context of binary quartic forms and their connections with pairs of ternary quadratic forms.

1. Definition and Explicit Construction

Let $$F(x,y) = a_4 x^4 + a_3 x^3 y + a_2 x^2 y^2 + a_1 x y^3 + a_0 y^4$$ be a binary quartic form over a field $K$ of characteristic zero. Classical invariant theory studies the action of $\operatorname{GL}_2(K)$ on space of such forms, focusing on quantities (invariants and covariants) that transform in specified ways under the group action.

The canonical degree six covariant of $F$ is defined by the determinant involving the derivatives of $F$ and its Hessian $H_F$: $$F_6(x,y) = \frac{1}{36} \begin{vmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \ \frac{\partial H_F}{\partial x} & \frac{\partial H_F}{\partial y} \end{vmatrix}.$$ Here, $H_F(x, y)$ is the Hessian of $F$, which is itself a quadratic covariant. The resulting $F_6(x, y)$ is a binary sextic (homogeneous of degree 6 in $x, y$) and is entirely determined by $F$.

This covariant is canonical: it arises without recourse to arbitrary normalization and is defined algebraically in terms of $F$ and its basic covariants.

2. Correspondence to Pair of Ternary Quadratic Forms

A principal achievement is a correspondence between the canonical degree six covariant of a binary quartic and a cubic covariant associated to pairs of ternary quadratic forms (Xiao, 5 Aug 2025). This proceeds as follows:

• Embedding binary quartics into pairs of ternary quadratic forms: The mapping $\varphi: V_4 \to W_4$ sends $F$ to a pair $(A_0, B_F)$, where $B_F$ is a symmetric $3 \times 3$ matrix constructed from the coefficients $a_i$ of $F$:

$$B_F = \begin{bmatrix} a_4 & a_3/2 & 0 \ a_3/2 & a_2 & a_1/2 \ 0 & a_1/2 & a_0 \end{bmatrix}.$$

• Cubic covariant construction: Given a pair of ternary quadratic forms, the canonical cubic covariant $C_3(u, v, w)$ is expressed as a Jacobian determinant of three associated quadratic covariants.

The central theorem (Theorem 1 in (Xiao, 5 Aug 2025)) establishes: $F_6(x, y) = C_3(x^2, xy, y^2).$ This means $F_6$ is obtained by evaluating the cubic covariant $C_3$ at $(x^2, x y, y^2)$, directly relating the invariant theory of binary quartics to that of pairs of ternary quadratics. This correspondence is not accidental; it encodes deep syzygetic and module-theoretic relationships between these algebraic objects.

3. Simultaneous Diagonalization and Matrix Interpretation

The degree six covariant not only encodes algebraic structure but directly informs the diagonalization of pairs of quadratic forms. Let $A$ and $B$ be $n \times n$ symmetric matrices over $K$ representing quadratic forms $f_A, f_B$, and define the associated binary form: $F_{A, B}(x, y) = \det(Ax - By).$ This polynomial factors as $\prod_{i=1}^n (s_i x - t_i y)$ over its splitting field $L$.

The paper demonstrates (Theorem 5) that:

• For each root $(s_i, t_i)$, the matrix $t_i A - s_i B$ has nullity one; its adjugate is rank one.
• The adjugate matrices yield vectors defining linear forms $\ell_i$.
• There exists an invertible matrix $U$ over $L$ such that

$$U^\top A U = \mathrm{diag}(s_1, \ldots, s_n), \quad U^\top B U = \mathrm{diag}(t_1, \ldots, t_n).$$

Thus, the construction of $F_{A, B}$ and its factorization provide canonical data from which diagonalizing transformations are built; furthermore, the process generalizes the algebraic role of the sextic covariant in the binary quartic case to a structure-theoretic approach for $n$-ary quadratic forms.

4. Diagonalizability Criterion Over Fields

A direct corollary (Corollary 6 in (Xiao, 5 Aug 2025)) of the above is a complete and effective criterion for simultaneous diagonalizability:

• A pair $(f_A, f_B)$ of $n$-ary quadratic forms over a field $K$ of characteristic zero is simultaneously diagonalizable over $K$ if and only if $F_{A, B}(x, y)$ splits completely into linear factors over $K$.

This reduces the algebraic complexity of diagonalizability—traditionally considered a subtle question about the structure of matrices and associated forms—to polynomial factorization.

5. Arithmetic and Invariant-Theoretic Significance

The canonical degree six covariant links to several important themes in arithmetic invariant theory:

• Discriminant-preserving embeddings: The covariant appears in the context of embedding binary quartic forms into spaces parameterizing quartic rings (see work by Matchett Wood) and is relevant for the computation and paper of discriminants in arithmetic statistics.
• Classical invariant theory: Its appearance as a minimal, natural construction involving the form and its Hessian makes it a fundamental component of the syzygetic relations among covariants of binary quartics.
• Parametrization and arithmetic statistics: The structure helps enable parametrization of algebraic objects such as quartic rings, facilitating precise counting results and distribution statements in algebraic number theory.
• Connections to higher covariants: The canonical degree of this covariant (six) is the minimal degree for a nontrivial "secondary" covariant derived from a binary quartic and its Hessian, reflecting the depth at which new algebraic relationships enter the invariant theory.

6. Broader Context and Generalizations

The construction and theory of the canonical degree six covariant demonstrate archetypal behavior for how classical invariant theory, linear algebraic properties, and arithmetic parametrizations interact in the paper of forms and their module-theoretic or geometric representations.

While specific to the case of binary quartics and ternary quadratics in the cited work (Xiao, 5 Aug 2025), the methods and structural correspondences generalize. The adjugate construction and the diagonalization approach admit extension to higher-degree forms and more general matrix tuples, with analogous canonical covariants arising at higher degrees and in more elaborate geometric settings.

A plausible implication is that analogous canonical covariants, constructed appropriately, should play central roles in the invariant theory and arithmetic of higher-degree forms and their associated moduli spaces.

7. Summary Table: Canonical Degree Six Covariant in Binary Quartic Context

Object	Definition/Construction	Role/Consequence
Binary quartic $F$	$a_4x^4 + a_3x^3y + a_2x^2y^2 + a_1xy^3 + a_0y^4$	Input form
Hessian $H_F$	$$\det \begin{bmatrix} \frac{\partial^2 F}{\partial x^2} & \frac{\partial^2 F}{\partial x \partial y} \ \frac{\partial^2 F}{\partial x \partial y} & \frac{\partial^2 F}{\partial y^2} \end{bmatrix}$$	Quadratic covariant
Degree six covariant $F_6$	$$(1/36) \cdot \det \begin{bmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \ \frac{\partial H_F}{\partial x} & \frac{\partial H_F}{\partial y} \end{bmatrix}$$	Encodes cubic covariants (links to ternary forms)
Pair $(A_0,B_F)$	Canonical image of $F$ as pair of ternary quadratic forms	Embeds quartic invariant data
Diagonalizability	$F_{A,B}$ splits into linears $\Leftrightarrow$ $A,B$ simultaneously diagonalizable	Algebraic criterion

In summary, the canonical degree six covariant is a central object in the invariant theory of binary quartics, precisely constructed from a binary quartic and its Hessian, with wide-ranging implications for simultaneous diagonalization of quadratic forms, explicit arithmetic criteria, and the broader connections between classical invariants and modern arithmetic geometry (Xiao, 5 Aug 2025).