---
title: 'Hermitian Polynomial: Definitions & Applications'
url: https://www.emergentmind.com/topics/hermitian-polynomial
type: topic
---

# Hermitian Polynomial: Definitions & Applications

Searching arXiv for recent and foundational papers on “Hermitian polynomial” across major mathematical usages.
to=arxiv_search tool code 的天天彩票{"query":"all:\"Hermitian polynomial\" OR all:\"Hermitian symmetric polynomial\" OR all:\"Hermitian polynomial filtering\" OR all:\"Hermitian matrix polynomials\"","max_results":10,"sort_by":"relevance"}
Relevant arXiv results include work on Hermitian polynomial filtering for eigenproblems, Hermitian symmetric polynomials and Hilbert’s 17th problem, Hermitian sums of squares, root counting with conjugate variables, noncommutative Hermitian squares, and matrix-polynomial synthesis for Hermitian operators.
A Hermitian polynomial is not a single universally fixed object. In current mathematical and computational literature, the term denotes several related but distinct constructions: a polynomial \(p(H)=\sum c_k H^k\) of a Hermitian matrix or operator; a Hermitian symmetric polynomial in complex variables with a Hermitian coefficient matrix; a self-adjoint element \(p=p^*\) in a free \(*\)-algebra; specialized univariate locator and evaluator polynomials in decoding Hermitian codes; and, in some matrix-theoretic settings, the characteristic polynomial of a Hermitian matrix [1512.08135] [1012.2479] [2109.00786] [0712.1775] [1405.5244]. The unifying theme is compatibility with an involution—complex conjugation, adjoint, or spectral symmetry—but the algebraic structure, positivity notion, and applications vary substantially from one context to another.

## 1. Terminological scope and defining structures

In numerical linear algebra, a Hermitian matrix \(A\) satisfies \(A=A^*\), has a real spectrum \(\lambda_1\ge \cdots \ge \lambda_n\), and admits an orthonormal eigenbasis. If \(p_m\) is a real polynomial, then \(p_m(A)\) is again Hermitian and diagonalizable in the same eigenbasis, with eigenvalues \(p_m(\lambda_i)\) obtained by pointwise application of the polynomial to the spectrum. In this setting, “Hermitian polynomial” refers to a polynomial filter \(p_m(A)\) designed to isolate prescribed spectral intervals [1512.08135].

In several complex variables, the standard object is a polynomial in \(z\) and \(\bar z\) whose coefficient matrix is Hermitian. Writing
\[
p(z,\bar z)=\sum_{\alpha,\beta} c_{\alpha\beta}\, z^\alpha \overline{z}^{\,\beta},
\]
Hermitian symmetry means \(c_{\alpha\beta}=\overline{c_{\beta\alpha}}\), equivalently that \(p\) is real-valued on \(\mathbb C^n\). In polarized form, bihomogeneous Hermitian symmetric polynomials correspond to Hermitian forms and admit signature invariants \((A,B)\) recording the numbers of positive and negative eigenvalues of the associated Hermitian matrix [1012.2479] [1003.0126].

In noncommutative algebra, one works in the free \(*\)-algebra \(\mathbb R\langle x_1,\dots,x_n\rangle\), where the involution fixes the generators and reverses words. A polynomial is Hermitian if \(p=p^*\). Under evaluation on self-adjoint operators or symmetric matrices, such a polynomial remains Hermitian, and positivity is formulated through operator inequalities and sums of Hermitian squares [2109.00786] [2503.12376].

Other usages are more specialized. In Hermitian code decoding, a “Hermitian polynomial” can mean a univariate error locator polynomial \(\Lambda(u)\) or error evaluator polynomial \(\Omega(u)\), obtained after reducing a bivariate Hermitian-curve decoding problem to a univariate one under semi-erasure decoding [0712.1775]. In random-matrix and spectral theory, the phrase may refer to the characteristic polynomial of a Hermitian matrix, whose zeros are real and whose coefficients encode traces, moments, and spectral constraints [1405.5244] [1708.00761].

## 2. Hermitian matrix polynomials and spectral filtering

For Hermitian eigenvalue problems, the spectral theorem gives
\[
A=X\Lambda X^*, \qquad p_m(A)=X\,p_m(\Lambda)\,X^*.
\]
This identity makes polynomial filtering effective: one chooses \(p_m(\lambda)\) to be close to \(1\) on a target interval \([a,b]\) and small outside, so that \(p_m(A)\) amplifies components in the invariant subspace associated with eigenvalues in \([a,b]\) and attenuates the others. In practice, the spectrum is first mapped to \([-1,1]\) by
\[
t(\lambda)=\frac{2\lambda-(\lambda_{\max}+\lambda_{\min})}{\lambda_{\max}-\lambda_{\min}},
\]
and the filter is expanded in Chebyshev polynomials, which can be applied matrix-free through the three-term recurrence [1512.08135].

A central implementation is the Thick-Restart Lanczos algorithm with deflation and polynomial filtering. After each restart, selected Ritz vectors are retained, converged vectors are locked, and the filtered operator \(B=(I-UU^\top)\rho_k(\widehat A)\) is applied through Chebyshev recurrences. The paper constructs filters by a least-squares approximation to a Dirac-\(\delta\) spike centered at \(\gamma\), uses Jackson or Lanczos \(\sigma\)-damping to reduce Gibbs oscillations, and chooses the degree \(k\) through a “bar” value \(\phi\) so that \(\rho_k(\xi)\) and \(\rho_k(\eta)\) fall below threshold. This supports spectrum slicing, in which different subintervals are treated independently and naturally in parallel [1512.08135].

More recent work replaces fixed-degree filters by adaptive Chebyshev step filters inside filtered subspace iteration. There, a partial degree \(k^{(i)}\) is selected at each iteration by monitoring the ratio of filtered Ritz values, and convergence is controlled by pointwise bounds for the step-function approximation in both undamped and damped settings. The same framework incorporates a spurious Ritz-value detection criterion based on the residual norm and the distance to the interval endpoints, and accelerates the dominant filtering step with MaSpMM. On the reported benchmarks, the method achieved average speedups of approximately \(14.5\times\) over EVSL-PSI and approximately \(2.14\times\) over CJ-FEAST [2604.00914].

The matrix-polynomial sense also appears in quantum algorithms. For a Hermitian contraction \(A\), the identity
\[
U=A+i\sqrt{I-A^2}, \qquad A=\frac12(U+U^\dagger)
\]
allows one to rewrite powers \(A^n\) as \(R_n(U)+R_n(U^\dagger)\), and hence any polynomial \(P(A)\) as \(\widetilde P(U)+\widetilde P(U^\dagger)\). This yields a block-encoding-free synthesis of Hermitian matrix polynomials using Generalized Quantum Signal Processing and postselection, with two ancillas and success probability
\[
\frac18\left\|\big(\widetilde P(U)+\widetilde P(U^\dagger)\big)\ket{\psi}\right\|^2
\]
for the desired branch [2512.18249].

## 3. Hermitian symmetric polynomials in complex variables

A Hermitian symmetric polynomial is a polynomial in complex variables and their conjugates whose coefficient matrix is Hermitian. Such a polynomial admits a holomorphic decomposition
\[
r(z,\bar z)=\sum_{j=1}^A |f_j(z)|^2-\sum_{k=1}^B |g_k(z)|^2=\|f(z)\|^2-\|g(z)\|^2,
\]
and the signature pair \(s(r)=(A,B)\) and rank \(A+B\) are basic invariants. The class \(P_0\) consists of squared norms, while \(P_k\) consists of those Hermitian symmetric polynomials for which \((r(z_i,\overline{z_j}))\) is positive semidefinite for every \(k\)-tuple of points. One has \(P_\infty=\bigcap_{k\ge 1}P_k=P_0\) [1012.2479].

The Hermitian analogue of Hilbert’s \(17\)-th problem asks when nonnegative Hermitian symmetric polynomials can be represented as quotients or divisors of squared norms. The positivity classes
\[
P_0 \subset Q \subset Q' \subset P_1
\]
are generally strict, but a central theorem establishes \(Q=Q'\): a nonnegative Hermitian symmetric polynomial divides a nonzero squared norm if and only if it is a quotient of squared norms [1012.2479]. This is a decisive difference from the real case, where a single square multiplier suffices after Artin.

Several explicit families illustrate the fine stratification of positivity. For
\[
r_\lambda(z,\bar z)=|z_1|^4+\lambda |z_1|^2|z_2|^2+|z_2|^4,
\]
one has \(r_\lambda\in P_1\) iff \(\lambda\ge -2\), \(r_\lambda\in P_0\) iff \(\lambda\ge 0\), and \(r_\lambda\in Q\) iff \(\lambda>-2\). Such examples show that nonnegativity does not coincide with being a squared norm, and that zero-set geometry imposes strong additional restrictions [1012.2479].

This theory is closely tied to CR geometry. Hermitian symmetric polynomials encode target hyperquadrics through their signature pairs, and products of indefinite Hermitian forms can exhibit striking rank collapse. Except for the trivial cases \((0,0)\), \((1,0)\), and \((0,1)\), every signature pair can arise as the signature of a product of two indefinite Hermitian symmetric polynomials [1003.0126]. Quillen’s Positivstellensatz supplies another foundational bridge: if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then for sufficiently large \(m\),
\[
s(z,\bar z)^m\,p(z,\bar z)=\sum_{k=1}^N |q_k(z)|^2,
\]
and an elementary proof can be given through the eventual positive-definiteness of an associated integral operator [1412.1570].

## 4. Noncommutative Hermitian polynomials and sums of Hermitian squares

In the free \(*\)-algebra \(\mathbb R\langle x\rangle\), Hermitian polynomials are the self-adjoint elements \(p=p^*\). If \(X=(X_1,\dots,X_n)\) is a tuple of self-adjoint operators or symmetric matrices, then \(p(X)\) is Hermitian whenever \(p\) is. The basic positivity cone is the set of sums of Hermitian squares
\[
\Sigma^2=\left\{\sum_{i=1}^\ell q_i^*q_i \mid q_i\in\mathbb R\langle x\rangle\right\},
\]
which evaluates to positive semidefinite operators under every Hermitian substitution [2109.00786].

This leads to noncommutative polynomial optimization. One minimizes a Hermitian nc polynomial \(p\) subject to Hermitian constraints \(g_j(X)\succeq 0\), and relaxes the problem through quadratic modules
\[
Q(g)=\left\{\sum p_i^* g_i p_i \mid g_i\in g\cup\{1\}\right\}.
\]
Under Archimedean assumptions, positivity on the operator semialgebraic set implies membership in \(Q(g)\), and moment/localizing matrix hierarchies yield convergent semidefinite programs. In the unconstrained case, matrix positivity is equivalent to membership in \(\Sigma\): \(f(X)\succeq 0\) for all symmetric \(X\) if and only if \(f\in\Sigma\) [2109.00786].

The same section of the literature also studies specific Hermitian polynomials arising from complete homogeneous symmetric polynomials. If \(H_{2d}\) denotes the fully symmetrized noncommutative lift of the even-degree complete homogeneous symmetric polynomial, then \(H_{2d}\) is a sum of \(\binom{n-1+d}{d}\) Hermitian squares, and this number is minimal. Moreover, for Hermitian operators \(X_1,\dots,X_n\),
\[
H_{2d}(X_1,\dots,X_n)\succeq \mu_{n,d}\big(X_1^{2d}+\cdots+X_n^{2d}\big),
\]
with an explicit best possible constant \(\mu_{n,d}\) [2503.12376]. The result is described as a noncommutative generalization of Hunter’s positivity theorem and produces new sum-of-squares representations even in the scalar commutative case.

A notable limitation is that positivity does not extend uniformly to arbitrary symmetrized Schur polynomials. The paper gives
\[
\sigma\!\big(s_{(2,2)}(x_1,x_2)\big)
\]
as a concrete example of a fully symmetrized nc polynomial that is not positive semidefinite on Hermitian matrix pairs, underscoring that complete homogeneous symmetric polynomials occupy a special position in the noncommutative theory [2503.12376].

## 5. Conjugate-variable, ideal-theoretic, and coding-theoretic usages

Another important usage concerns generalized polynomials in a complex variable and its conjugate. Writing
\[
f(z,\bar z)=\sum_{i,j} a_{ij} z^i\bar z^j,
\]
one may regard \(f(z,w)\in\mathbb C[z,w]\) and impose \(w=\bar z\) only at evaluation. For systems with conjugate variables, the Hermitian Killing form
\[
K_\xi([f],[g])=\mathrm{Tr}(M_\xi M_f M_{g^*})
\]
is a Hermitian sesquilinear form on the quotient algebra \(\mathbb C[z,w]/I\), where \(I\) is generated by the mixed-variable equations and their conjugates. Its signature counts “conjugated singles,” that is, true zeros \(z\in\mathbb C\), and yields new bounds for harmonic polynomials \(h(z,\bar z)=p(z)+\overline{q(z)}\). In particular, if \(\deg p=n\), \(\deg q=m\), and \(n-2>m\), then \(h(z,\bar z)=0\) admits at most \(n^2-1\) or \(n^2-2\) solutions depending on whether \((n-1)a_{n-1}-2n a_{n-2}\) vanishes [2406.15628].

Hermitian ideals provide a related but distinct framework. In \(\mathbb C[z,\bar z]\), a Hermitian polynomial is fixed by the involution \(f\mapsto \overline{f}\), and one asks when it can be written as a Hermitian sum of squares modulo a Hermitian ideal \(I\). For the one-variable ideals \(I=(z^N\bar z^N-1)\), matrix positivity conditions on orbit Gram matrices are not only necessary but sufficient, and are equivalent to the positivity of an associated block Toeplitz matrix and to an operator-valued Riesz–Fejér factorization [2012.03450]. This is a concrete Hermitian Positivstellensatz on a specific algebraic variety.

In coding theory, the terminology changes again. For Hermitian codes over \(\mathbb F_{q^2}\), a “Hermitian polynomial” refers to the univariate error locator polynomial \(\Lambda(u)\) and error evaluator polynomial \(\Omega(u)\) used after semi-erasure reduction. The central gain is that a bivariate search over \(q^3\) affine points becomes a univariate Chien-like search over \(q^2\) points, and the Reed–Solomon Forney formula carries over directly:
\[
E_i=-\frac{\Omega(u_i)}{\Lambda'(u_i)}.
\]
The paper explicitly notes that this meaning is unrelated to Hermite orthogonal polynomials from analysis [0712.1775].

## 6. Characteristic polynomials of Hermitian matrices and related spectral constructions

In several matrix-theoretic papers, “Hermitian polynomial” means the characteristic polynomial of a Hermitian matrix. Because Hermitian spectra are real, these polynomials have real zeros, and their averages under Hermitian matrix diffusion satisfy exact diffusion-type equations:
\[
\partial_\tau \pi_N(z,\tau)=-\frac{1}{2N}\partial_z^2 \pi_N(z,\tau), \qquad
\partial_\tau \theta_N(z,\tau)=+\frac{1}{2N}\partial_z^2 \theta_N(z,\tau).
\]
Their integral representations lead to Airy asymptotics at soft edges and Pearcey asymptotics at cusp-merging points, while the logarithmic derivative of the averaged characteristic polynomial obeys a viscous Burgers equation [1405.5244].

Arithmetic restrictions become prominent when the Hermitian matrix entries are roots of unity. If \(H\in H_n(q)\) is a Hermitian matrix with entries in \(C_q\), then
\[
\chi_H(x)\in \mathbb Z[\zeta+\zeta^{-1}][x],
\]
and the residue classes of \(\chi_H\) modulo powers of \(1-\zeta\) satisfy explicit upper bounds. For \(q=p^f\) with \(p\) odd prime, for example,
\[
|\mathcal X_n(p^f,e)|\le p^{(e-1)^2},
\]
while two-adic cases exhibit additional parity-dependent structure [2106.05477].

Characteristic polynomial coefficients also encode refined spectral data. Using traces \(s_k=\mathrm{tr}(H^k)\) and Hankel determinants, one can reconstruct the minimal polynomial, factor the characteristic polynomial by grouping eigenvalues with the same multiplicity, and compute monotone rational approximations to the minimal eigenvalue gap and to the extremal eigenvalues. The same symbolic framework yields invariants for classifying unitary orbits of Hermitian operators [1708.00761].

For cubic polynomials, the almost-companion-matrix approach shows that a monic cubic is the characteristic polynomial of a Hermitian \(3\times 3\) matrix if and only if its coefficients are real and the depressed discriminant satisfies \(\Delta(p,q)\le 0\). The same paper gives a complete coefficient characterization for cubic unitary ACMs and uses structured Hermitian or unitary realizations as an alternative route to root finding [2301.07435].

A different but related tradition derives orthogonal polynomials of a discrete variable from Hermitian Jacobi matrices. There, “Hermitian” refers to real symmetric tridiagonal operators \(H\) whose eigenvalue problem produces orthogonal polynomial systems through factorization, shape invariance, and exact Heisenberg evolution. The resulting polynomials are not classical Hermite polynomials; rather, they belong to discrete families such as Krawtchouk, Hahn, Racah, Meixner, Charlier, and their \(q\)-analogues [0712.4106].

Source: https://www.emergentmind.com/topics/hermitian-polynomial