Degree-Preserving Sum-of-Squares Factorization

• Degree-preserving SOS factorization is a set of techniques for representing nonnegative structured polynomials as Hermitian squares without increasing the original degree.
• It employs methods such as Fejér–Riesz factorization, Gram matrix constructions, and hybrid numeric-symbolic algorithms to achieve exact certificates with Gaussian rational data.
• Applications span discrete-time signal processing, SDP-based optimization, and noncommutative settings, while addressing challenges in ensuring exact arithmetic and degree bounds.

Degree-preserving sum-of-squares spectral factorization denotes a class of representations in which a nonnegative trigonometric polynomial, spectral density, matrix polynomial, or related structured object is written as a Hermitian square, a weighted sum of Hermitian squares, or a Gram-type sum-of-squares certificate without increasing the relevant degree parameter. In the univariate trigonometric setting, the prototype is the Fejér–Riesz factorization $f=Q Q^\star$, where $\deg Q$ matches the trigonometric degree of $f$; in later developments, the same degree-preserving principle appears in spectrahedral lifts for Newton-type inequalities, in stochastically minimal factorizations of rational discrete-time spectra, and in noncommutative and pseudomoment constructions where the factors or Gram vectors retain the original degree bound (Magron et al., 2022, Kummer, 2020, Baggio et al., 2014, Klep et al., 12 Nov 2025).

1. Core definitions and the meaning of degree preservation

For trigonometric univariate polynomials with Gaussian integer coefficients, one works with Hermitian Laurent polynomials

$$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$

where $f_0\in\mathbb R$, $f_k\in\mathbb C$, and $f\in \mathscr H(\mathbb Z)[z]$ means that all $f_k$ lie in $\mathbb Z[i]$. Positivity means $f(z)\ge 0$ for all $\deg Q$0. A sum of Hermitian squares (SOHS) decomposition has the form

$\deg Q$1

and a weighted SOHS allows nonnegative weights on the unit circle, with the algorithms in the Gaussian-coefficient setting using constant positive rationals or moduli of Gaussian numbers (Magron et al., 2022).

In this setting, “degree-preserving” means that if $\deg Q$2, then each output factor $\deg Q$3 has degree at most $\deg Q$4. For a single Fejér–Riesz factor, the preservation is exact: the spectral factor has the same trigonometric degree as the input. For Gram-matrix constructions in algebraic SOS, the same phrase refers to a certificate whose building blocks already have the degree required so that squaring reproduces the target degree and no higher-degree auxiliary terms are introduced. The literature therefore uses one expression for two closely related phenomena: preservation of harmonic order in spectral factorization, and preservation of polynomial degree in Gram/SOS certificates (Magron et al., 2022, Kummer, 2020).

A persistent source of confusion is the distinction between the degree of each factor and the number of squares. Degree preservation does not imply a single-square factorization over the coefficient ring of interest. In particular, exactness over $\deg Q$5 can force a passage from one Hermitian square over $\deg Q$6 to several weighted Hermitian squares over Gaussian rationals, even though the degree bound on each factor remains unchanged (Magron et al., 2022).

2. Fejér–Riesz factorization and exact SOHS over Gaussian coefficients

The classical univariate statement is the Riesz–Fejér theorem: if

$\deg Q$7

is real-valued and nonnegative on the unit circle, then there exists a complex polynomial $\deg Q$8 of degree $\deg Q$9 such that

$f$0

Equivalently, in Laurent form on $f$1, one has $f$2. The minimum-phase version places all zeros of $f$3 inside the open unit disk. This is the basic degree-preserving spectral factorization theorem for trigonometric polynomials (Magron et al., 2022).

For Gaussian integer coefficients, however, exact arithmetic introduces a second layer of difficulty. Although positivity on the unit circle guarantees a single Hermitian square over $f$4, the factor $f$5 generally has complex algebraic coefficients rather than coefficients in $f$6. The exact-certification problem therefore changes character: one no longer asks merely for existence of a complex spectral factor, but for an exact certificate with Gaussian rational data. The solution developed for this setting is to allow either perturbation-compensation or rounding-projection, yielding exact weighted SOHS decompositions with Gaussian rational coefficients while preserving the degree bound on every factor (Magron et al., 2022).

The perturbation idea is explicit. For $f$7 positive on the unit circle, there exists $f$8 such that with $f$9 one has $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$0 on the unit circle, where $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$1 is the maximum bitsize of the coefficients. One then factorizes or Gram-represents the perturbed polynomial numerically and compensates the exact remainder symbolically. This converts approximate numerical data into an exact certificate over Gaussian rationals. The resulting decomposition may contain several Hermitian squares, but each factor still has degree at most $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$2 (Magron et al., 2022).

This exactness issue clarifies a broader misconception. Fejér–Riesz solves positivity over $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$3; it does not by itself solve exact certification over arithmetic coefficient domains. The latter requires hybrid numeric-symbolic procedures, coefficient reconstruction, and explicit error control.

3. Hybrid numeric-symbolic algorithms

Three algorithms are developed for exact weighted SOHS decompositions of positive trigonometric univariate polynomials with Gaussian coefficients: a root-isolation method csos1, an SDP Gram-matrix method csos2, and a Peyrl–Parrilo-style rounding-projection method csos3. All three return exact certificates with Gaussian rational coefficients, all preserve degree, and all admit polynomial bit-complexity and output-size bounds in the degree and coefficient bitsize (Magron et al., 2022).

Algorithm	Main numerical ingredient	Complexity / output size
csos1	Complex root isolation plus compensation	$$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$4 bit operations; output bitsize $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$5
csos2	Complex SDP Gram matrix plus Cholesky and compensation	$$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$6 bit operations; output bitsize $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$7
csos3	Complex SDP plus rounding, projection, and $$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$8 factorization	$$f(z)=f_0+\sum_{k=1}^{d}\left(\frac{f_k}{z^k}+\overline{f_k}\,z^k\right),$$9 bit operations; output bitsize $f_0\in\mathbb R$0

The root-isolation algorithm csos1 starts from the perturbed polynomial $f_0\in\mathbb R$1, forms the algebraic polynomial $f_0\in\mathbb R$2 of degree $f_0\in\mathbb R$3, isolates roots with sufficient precision, builds a spectral core

$f_0\in\mathbb R$4

and then compensates the exact remainder using rank-1 Hermitian squares. The stopping criterion is

$f_0\in\mathbb R$5

where $f_0\in\mathbb R$6 is the Hermitian remainder. Once this inequality holds, the right-hand side is an exact SOHS with Gaussian rational coefficients, consisting of the spectral core together with linear rank-1 terms, and all factors have degree at most $f_0\in\mathbb R$7 (Magron et al., 2022).

The SDP method csos2 formulates positivity as a Toeplitz-constrained Hermitian PSD Gram problem. With

$f_0\in\mathbb R$8

one seeks $f_0\in\mathbb R$9 such that

$f_k\in\mathbb C$0

with coefficient-matching constraints expressed by Toeplitz selector matrices. The SDP maximizes a lower bound on the smallest eigenvalue of $f_k\in\mathbb C$1, which improves numerical robustness. If $f_k\in\mathbb C$2 is rank one, a single Hermitian square is recovered; otherwise, finite-precision Cholesky produces approximate squares that are made exact by the same compensation mechanism used in csos1 (Magron et al., 2022).

The rounding-projection method csos3 removes the perturbation and instead enforces exact coefficient matching directly. After solving the complex SDP for $f_k\in\mathbb C$3, one rounds the numerical Gram matrix to Gaussian rationals, projects it onto the affine space defined by the coefficient constraints, and factors the exact projected matrix via $f_k\in\mathbb C$4. If the diagonal entries of $f_k\in\mathbb C$5 are positive rationals, the output is an exact weighted SOHS

$f_k\in\mathbb C$6

with $f_k\in\mathbb C$7 rational and each $f_k\in\mathbb C$8 of degree at most $f_k\in\mathbb C$9 (Magron et al., 2022).

The empirical comparison in the same work reports that for the random family

$f\in \mathscr H(\mathbb Z)[z]$0

which is positive on the unit circle, csos1 runs markedly faster than csos2 and csos3, consistent with the theoretical bounds. The more expensive SDP-based route remains important because it supports constrained synthesis tasks; in particular, the rounding-projection machinery is used to design a certified linear-phase finite impulse response filter under passband and stopband constraints (Magron et al., 2022).

4. Spectral LMIs, Gram matrices, and degree-preserving SOS identities

A second major line of work treats degree-preserving SOS spectral factorization through spectrahedral lifts rather than through one-variable Fejér–Riesz factorization. The basic object is a symmetric spectral convex set

$f\in \mathscr H(\mathbb Z)[z]$1

where $f\in \mathscr H(\mathbb Z)[z]$2 is described by a linear matrix inequality and $f\in \mathscr H(\mathbb Z)[z]$3 denotes the eigenvalue vector. Under a representation-theoretic “short representation” assumption, such spectral LMIs lift to actual spectrahedra: there exists an $f\in \mathscr H(\mathbb Z)[z]$4-linear map

$f\in \mathscr H(\mathbb Z)[z]$5

such that $f\in \mathscr H(\mathbb Z)[z]$6 (Kummer, 2020).

This lift yields constructive SOS certificates for hyperbolic-polynomial inequalities. A central example is the normalized Newton gap

$f\in \mathscr H(\mathbb Z)[z]$7

which is proved to be a sum of squares. The matrix analogue, obtained by replacing $f\in \mathscr H(\mathbb Z)[z]$8 with $f\in \mathscr H(\mathbb Z)[z]$9, is also SOS in the entries of $f_k$0. The Gram matrix has the explicit form

$f_k$1

and the associated identity is written as $f_k$2, where $f_k$3 lists a basis of $f_k$4 minors (Kummer, 2020).

The degree-preserving content is explicit. In the diagonal case, one may use the vector $f_k$5 of square-free monomials of degree $f_k$6; in the matrix case, one uses minors of degree $f_k$7. In either case, the Gram vectors already have degree $f_k$8, so squaring them yields degree $f_k$9, exactly matching the target Newton-gap polynomial. The construction therefore preserves degree in the same structural sense as Fejér–Riesz: the certificate does not pay for positivity by increasing the degree of the factors (Kummer, 2020).

The same spectrahedral machinery also yields SOS forms for strengthened correlation inequalities. For Renegar derivatives $\mathbb Z[i]$0, the quantity

$\mathbb Z[i]$1

is a sum of squares whenever $\mathbb Z[i]$2 lies in the relevant hyperbolicity cone and $\mathbb Z[i]$3. This situates degree-preserving SOS factorization within the broader theory of derivative relaxations of the PSD cone and hyperbolic programming (Kummer, 2020).

The framework is not universal. The paper proves that $\mathbb Z[i]$4 is spectrahedral when the defining $\mathbb Z[i]$5-representation is short, but explicitly leaves open the general statement that every symmetric spectrahedral cone should have a spectrahedral spectral lift. It also notes that degree preservation of the Gram construction does not by itself settle optimality of SOS length or minimal Gram size (Kummer, 2020).

5. Matrix-valued, rational, and noncommutative extensions

For matrix-valued rational discrete-time spectral densities, the relevant factorization problem is no longer scalar Fejér–Riesz but para-Hermitian spectral factorization. A rational spectrum $\mathbb Z[i]$6 is a para-Hermitian matrix positive semidefinite on the unit circle, and the main constructive theorem states that for any spectrum of normal rank $\mathbb Z[i]$7 and any unmixed-symplectic analyticity regions $\mathbb Z[i]$8, there exists

$\mathbb Z[i]$9

such that

$f(z)\ge 0$0

$f(z)\ge 0$1 is analytic in $f(z)\ge 0$2, its right inverse is analytic in $f(z)\ge 0$3, and $f(z)\ge 0$4 is stochastically minimal: $f(z)\ge 0$5

When $f(z)\ge 0$6, the factor is unique up to a constant orthogonal left multiplier. For Laurent-polynomial spectra, this reduces to a matrix Fejér–Riesz statement, and in the polynomial case the degree relation is $f(z)\ge 0$7 in the Laurent sense (Baggio et al., 2014).

The constructive proof proceeds through Smith–McMillan reduction, separation of poles and zeros by diagonal canonical factors, reduction to an $f(z)\ge 0$8-polynomial para-Hermitian positive definite matrix, and an iterative degree-reduction algorithm that repeatedly lowers highest column degrees until a constant matrix remains, at which point a Cholesky factorization completes the construction. The resulting factor is not merely an existence statement; it is a degree-minimal or, more precisely, stochastically minimal spectral factor with analyticity regions selected by symplectic-type conditions (Baggio et al., 2014).

A different extension appears in matrix-valued noncommutative trigonometric polynomials. Let $f(z)\ge 0$9 be either the free semigroup $\deg Q$00 or the free product group $\deg Q$01, and let $\deg Q$02 be a discrete group. If

$\deg Q$03

is a matrix-valued trigonometric polynomial of $\deg Q$04-degree at most $\deg Q$05 and is uniformly strictly positive on all unitary representations, then there exists an analytic polynomial $\deg Q$06 such that

$\deg Q$07

with $\deg Q$08. The $\deg Q$09-degree bound is optimal, and strict positivity is essential in the general theorem (Klep et al., 12 Nov 2025).

The proof combines several ingredients: shortlex order on $\deg Q$10, a positive-semidefinite Parrott theorem with entries given by functions on a group, PSD matrix-completion results for $\deg Q$11 and $\deg Q$12, realization of PSD kernels by unitary representations, and an operator-system/CP-map route using Arveson extension and Choi factorization. The degree-preserving content is built into the shortlex truncation: the analytic factor never exceeds the original $\deg Q$13-degree. In the special case of trivial $\deg Q$14, the same work proves a “perfect” Positivstellensatz on $\deg Q$15 that does not require strict positivity; the result is sharp, as shown by counterexamples for $\deg Q$16 and $\deg Q$17 (Klep et al., 12 Nov 2025).

These matrix and noncommutative developments broaden the scope of degree-preserving spectral factorization without changing its structural aim. The factor remains analytic in the appropriate sense, positivity is encoded by $\deg Q$18 or $\deg Q$19, and the degree or McMillan invariant of the factor is fixed by the degree of the original spectral object.

6. Applications, reinterpretations, and unresolved issues

In discrete-time signal processing, nonnegative trigonometric polynomials serve as power spectra, and degree-preserving spectral factorization produces stable or minimum-phase finite impulse response filters through a factor $\deg Q$20 satisfying $\deg Q$21. The exact SOHS algorithms over Gaussian coefficients were used to design a certified linear-phase FIR filter by minimizing stopband energy subject to passband and stopband constraints, with exact positivity certification recovered after numerical optimization by rounding-projection and rational reconstruction (Magron et al., 2022).

The literature also uses closely related degree-preserving spectral/SOS ideas in computational complexity. For a broad class of planted problems, eigenvalues of degree-$\deg Q$22 matrix polynomials in the instance variables are shown to be as powerful as SoS semidefinite programs of roughly degree $\deg Q$23. In tensor PCA, a degree-2 matrix polynomial detects at the scale $\deg Q$24, matching degree-4 SoS; for sparse PCA, the lower bounds indicate that degree-$\deg Q$25 SoS cannot beat the PCA and diagonal-threshold regimes by polynomial factors, suggesting that further improvement may require sub-exponential time (Hopkins et al., 2017). Here “spectral factorization” no longer refers to a trigonometric power spectrum, but the same design principle remains: a low-degree Gram- or matrix-polynomial object captures the SoS certificate without increasing degree.

An even closer analogue appears in positivity-preserving extensions of pseudomoments over the hypercube. Starting from a degree-2 pseudomoment matrix $\deg Q$26, the higher-degree pseudomoment matrix is organized in a multiharmonic basis with Gram blocks

$\deg Q$27

Pseudomoment values are given by forest sums with Möbius coefficients,

$\deg Q$28

and the central lifting theorem states that if

$\deg Q$29

then the construction yields a valid degree-$\deg Q$30 pseudoexpectation. The same framework extends Sherrington–Kirkpatrick lower bounds from degree 4 to degree 6 and gives degree $\deg Q$31 extensions for random high-rank projection matrices (Kunisky, 2020).

Several limitations recur across these strands. A single square over $\deg Q$32 need not yield an exact certificate over $\deg Q$33; strict positivity is essential in the general noncommutative theorem; the fully general spectral-LMI lifting problem remains open beyond short representations; the univariate Gaussian-coefficient work leaves the case of vanishing on the unit circle for future study; and in the hypercube setting the low-rank case appears to require a next-order correction beyond the leading multiharmonic Gram picture (Magron et al., 2022, Klep et al., 12 Nov 2025, Kunisky, 2020). These constraints do not weaken the core paradigm. Rather, they delimit the regimes in which degree preservation can coexist with exact arithmetic, analytic structure, and constructive positivity certificates.