# Matrix Polynomial Certification: Scope and Prior Machinery

The multicomponent extension naturally produces matrix stability conditions. The final result is deliberately split into a proved collapse theorem and classical terminal-certification machinery.

## Proved simplex sparse result

For independent simplex coordinates `y=(Y_1,...,Y_r)`, `Y_0=1-sum Y_i`, consider

`h(p,s,y)=h_0(p,s)+sum_i y_i r_i(p)+phi(y)`, 

where `h_0 in Q_43`, each `r_i` is a pressure polynomial of degree at most four, and `phi(y)=1/2 y^T A y+b^T y+c` with constant `A` positive definite. The acoustic primitive fields `h_p,-h_pp,h_ppp` are affine in barycentric composition coordinates, while `h_s,h_ss` are composition independent and the composition block of the `(s,y)` stability Hessian is constant. Hence scalar safety reduces exactly to the `r+1` simplex vertices, and matrix stability reduces to one constant positive-definite test on `A`.

## Exact no-go outside the sparse class

The vertex property is not valid for a generic function that is merely multiaffine in independent coordinates restricted to a simplex. For example `1/10-Y_1Y_2` is positive at the three vertices of the two-dimensional composition simplex but negative at `(1/2,1/2)`. The release embeds this function into `h_ppp` while maintaining strict volume, compressibility, temperature, entropy curvature, and a uniformly positive-definite composition Hessian. Thus naive simplex-vertex certification of generic full tensor composition dependence is false.

## Classical matrix-polynomial terminal certificates

If an independent dimension-collapse argument reduces a stability matrix to a univariate symmetric polynomial matrix, established results are available. Positive semidefinite univariate matrix polynomials admit sum-of-squares/factorization descriptions; matrix Fejer-Riesz theory and later real algebraic results provide exact structural tools. Positivstellensatz and rational-SOS representations also exist for polynomial matrices on intervals and semialgebraic sets.

These are classical ingredients. This release does **not** claim a new general matrix Markov-Lukacs theorem, nor does it implement a complete rational matrix-SOS backend. The EOS-specific contribution is the sparse simplex reduction that can expose such lower-dimensional matrix problems.

Selected prior work:

- C. Hanselka and R. Sinn, *Positive Semidefinite Univariate Matrix Polynomials*, Mathematische Zeitschrift 292 (2019), 83-101, DOI 10.1007/s00209-018-2137-7.
- T.-H. Le and N.-T. Pham, sum-of-squares-of-rational-function representations for PSD polynomial matrices on intervals, arXiv:1901.02360.
- S. Gift and H. J. Woerdeman, *Real factorization of positive semidefinite matrix polynomials*, Linear Algebra and its Applications (2024), DOI 10.1016/j.laa.2023.12.005.