Denominator-SOS Hierarchy

• The Denominator-SOS Hierarchy is a framework that represents nonnegative polynomials as ratios of sums of squares with explicit control over the denominator degree.
• It leverages semidefinite programming and moment relaxations to provide actionable certificates and delineate representability barriers in polynomial optimization.
• Applications include certifying impossibility results for symmetric sextics and Motzkin’s sextic, thereby extending classical Positivstellensatz methods.

The Denominator-SOS (Sum of Squares) hierarchy is a structured framework for characterizing and certifying representations of nonnegative polynomials and rational functions as ratios of sums of squares (SOS) polynomials, with explicit control over the degree and structure of the denominator. This hierarchy arises from the interplay between real algebraic geometry, semidefinite programming (SDP), and the Positivstellensatz, addressing foundational questions regarding the impossibility (or necessity) of Hilbert–Artin representations with prescribed denominator characteristics. Its study elucidates the fine gradation of representability barriers among nonnegative functions, and provides algorithmic certificates delimiting the boundaries of such representations (Guo et al., 2012, Huang et al., 2023).

1. Definitions and the Hierarchy Structure

Let $K \subseteq \mathbb{R}$ (e.g., $K = \mathbb{Q}$) and $f(x) \in K(x_1, \dots, x_n)$ be a rational function. A Hilbert–Artin representation is a decomposition

$$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$

where $u_i, v_j \in K[x_1, \dots, x_n]$. By Artin’s solution to Hilbert’s 17th problem, every $f \geq 0$ admits some such representation for sufficiently large denominator degree.

Define, for $d = 0, 1, 2, \dots$,

$$H_d := \big\{ f : \exists~\mathrm{Hilbert\text{–}Artin~representation~with}~\max \deg v_j \leq d \big\},$$

producing an ascending chain $H_0 \subseteq H_1 \subseteq H_2 \subseteq \cdots$. Here $H_0$ is the cone of SOS polynomials, $K = \mathbb{Q}$0 extends to denominators of degree $K = \mathbb{Q}$1, and so forth, with the union encompassing all nonnegative $K = \mathbb{Q}$2.

Denominator-SOS hierarchies are similarly defined in the context of polynomial optimization over basic semialgebraic sets $K = \mathbb{Q}$3 involving equations $K = \mathbb{Q}$4 and inequalities $K = \mathbb{Q}$5, modeling generalized moment problems and their dual SOS relations (Guo et al., 2012, Huang et al., 2023).

2. Theoretical Foundations

The heart of the Denominator-SOS hierarchy lies in characterizing which nonnegative functions admit a Hilbert–Artin representation with denominators of limited structure or degree, and providing certificates when such representation is impossible.

• SDP Reformulation: The existence of a denominator-SOS representation of degree $K = \mathbb{Q}$6 manifests as a system of linear matrix inequalities over Gram matrices associated with possible numerator and denominator monomials. For $K = \mathbb{Q}$7 with prescribed degree bounds, feasibility becomes an SDP problem in the Gram matrix variables.
• Farkas’ Lemma for SDPs: A generalization of Farkas’ Lemma (Alizadeh 1993) establishes that infeasibility of the SDP is certified by a dual vector $K = \mathbb{Q}$8 such that certain moment and localizing matrices satisfy $K = \mathbb{Q}$9, with $f(x) \in K(x_1, \dots, x_n)$0, furnishing an exact impossibility certificate.
• Exact Rational Certificates: Numerical interior-point solutions can be post-processed—by convex combination and rational rounding—to yield an exact rational certificate $f(x) \in K(x_1, \dots, x_n)$1, which gives rise to a separating linear form $f(x) \in K(x_1, \dots, x_n)$2 demonstrating $f(x) \in K(x_1, \dots, x_n)$3 (Guo et al., 2012).

The extension of the Denominator-SOS concept to polynomial optimization over semialgebraic sets $f(x) \in K(x_1, \dots, x_n)$4 is accomplished by defining the “denominator-augmented” quadratic module, e.g.,

$f(x) \in K(x_1, \dots, x_n)$5

which enlarges the feasible cone and overcomes archimedean and radicality barriers (Huang et al., 2023).

3. Algorithmic Realization

The verification of nonexistence of SOS denominator representations with specified degree proceeds via the following algorithmic schema (Guo et al., 2012):

1. Big-M Regularization: Apply an SDP with additional trace/eigenvalue bounds to enforce strict feasibility.
2. Interior-Point and Rational Recovery: Numerically solve the regularized primal/dual SDP to high precision. Locate a strictly feasible point, then form a convex combination to ensure the dual objective remains negative. Coordinates are rounded to rationals to yield a valid dual certificate.
3. Certificate Validation: The resulting rational $f(x) \in K(x_1, \dots, x_n)$6 is a separating form, certifying that for all candidate denominator SOS polynomials within degree constraints, no admissible representation exists.

In the polynomial optimization and moment-SOS context, the denominator-SOS relaxations take the form

$f(x) \in K(x_1, \dots, x_n)$7

where nonnegative $f(x) \in K(x_1, \dots, x_n)$8, $f(x) \in K(x_1, \dots, x_n)$9 are SOS, and the dual moment relaxation includes corresponding moment and localizing matrix positivity constraints. Feasibility and flat extension of the obtained truncated moment matrices are then checked to certify global optimality (Huang et al., 2023).

4. Applications and Concrete Results

The Denominator-SOS hierarchy provides a systematic method for proving impossibility results for SOS denominator representations of definite polynomials. Key examples include:

• Symmetric Sextics: For the family $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$0, explicit rational certificates demonstrate, for various $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$1, the nonexistence of denominator-SOS representations below specified degrees. For instance:
  • $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$2 (certificate of size $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$3),
  • $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$4 ($$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$5),
  • $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$6 ($$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$7),
  • and similar exclusions for $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$8, $$f(x) = \frac{N(x)}{D(x)}, \quad N(x) = \sum_{i=1}^L u_i(x)^2,\quad D(x) = \sum_{j=1}^{L'} v_j(x)^2,$$9, $u_i, v_j \in K[x_1, \dots, x_n]$0 (Guo et al., 2012).
• Non-uniform and Restricted-Term Denominators: Unlike earlier results requiring uniform denominators (e.g., powers of $u_i, v_j \in K[x_1, \dots, x_n]$1 as in Reznick’s bounds), the method applies to arbitrary non-uniform SOS sums or denominators with specified term restrictions, such as Landau’s reduction omitting a variable (Guo et al., 2012).
• Classical Counterexamples: Motzkin’s sextic polynomial is shown to evade $u_i, v_j \in K[x_1, \dots, x_n]$2; certain rational functions (e.g., Motzkin divided by $u_i, v_j \in K[x_1, \dots, x_n]$3) are excluded from $u_i, v_j \in K[x_1, \dots, x_n]$4.
• Moment-SOS Hierarchies with Denominators: The extension to the polynomial optimization framework allows for finite convergence guarantees even in the presence of non-real radical constraints or non-archimedean sets, via denominator-augmented relaxations (Huang et al., 2023).

5. Implications and Limitations

The Denominator-SOS hierarchy clarifies the gradations between the existence and nonexistence of low-degree denominator SOS representations. The explicit construction of impossibility certificates fills in “no-barrier-below-d” cells in the hierarchy for various families of polynomials, particularly symmetric forms where classical results (such as Reznick’s uniform denominator bounds) do not exclude non-uniform, lower-degree denominators (Guo et al., 2012).

A table summarizing example results:

Instance	Minimum Denominator Degree Excluded	Certificate Size (m)
$u_i, v_j \in K[x_1, \dots, x_n]$5	2	369
$u_i, v_j \in K[x_1, \dots, x_n]$6	4	2751
Motzkin (sextic)	0 ($u_i, v_j \in K[x_1, \dots, x_n]$7)	Not specified

A plausible implication is that non-uniform denominators often do not enable lower-degree representations for extremal examples, and explicit certification is now feasible for such claims.

6. Connections, Extensions, and Open Questions

The Denominator-SOS hierarchy interacts deeply with several core areas:

• Polynomial Optimization: In contexts where the standard quadratic module is not archimedean or the constraint ideal is not real-radical, classical SOS or moment relaxations may fail to attain exactness. The introduction of denominators, possibly as sphere weights $u_i, v_j \in K[x_1, \dots, x_n]$8 or powers of constraint polynomials $u_i, v_j \in K[x_1, \dots, x_n]$9, enables the restoration of finite convergence and strong duality under mild assumptions, provided suitable KKT conditions hold (Huang et al., 2023).
• Finite Convergence: It is established that when denominator powers are sufficiently large and $f \geq 0$0 is taken large enough, the denominator-SOS relaxations and their dual moment programs yield exactness, with atomic measures corresponding to global minimizers upon satisfaction of the flat extension condition.
• Symbolic vs. Numeric Certification: While the method is fundamentally numerical (SDP/interior-point), it produces rational certificates guaranteeing mathematical correctness. However, the computational size of the SDP (and resulting certificates) can be limiting. Whether explicit symbolic (algebraic) bounds or characterizations can supplant SDP-based infeasibility certificates remains open.
• Future Directions: Characterizing precisely which polynomials admit minimal-degree denominator SOS Hilbert–Artin representations, extending the hierarchy to infinite-dimensional contexts (e.g., trigonometric polynomials), and exploring decision complexity or tightness (i.e., whether every extremal case genuinely requires a non-uniform denominator) are significant open questions (Guo et al., 2012).

7. Summary and Significance

The Denominator-SOS hierarchy transforms the classical existential theory of Hilbert and Artin into an explicit obstruction-theoretic framework with concrete separation certificates. Through semidefinite programming and rational post-processing, one can demarcate the exact locations where low-degree denominator-SOS representations are impossible for nonnegative polynomials and rational functions. The methodology extends naturally to generic polynomial optimization problems on semialgebraic sets, removing obstructions arising from nonarchimedean or non-real-radical constraints by the systematic use of denominators. This hierarchy thus critically refines the structure of nonnegativity certificates in real algebraic geometry, advancing both the theoretical landscape and the constructive power of SDP-based proof systems (Guo et al., 2012, Huang et al., 2023).