# Verification report — AMR-036-0044 (Eremenko: generic stabilizability by static output feedback, n = mp)

Verification date: 2026-10-03. Paper: "Open Sets of Linear Systems with n = mp That Are Not Stabilizable by Static
Output Feedback: New Cases of a Problem of Eremenko" (14 pages).

**Verdict.** Partial result; the problem stays open. The question asks for which (m, p) the generic real system
x' = Ax + Bu, y = Cx with m outputs, p inputs and n = mp states is stabilizable by a real static output feedback
u = Ky. The note settles new cases, all negatively: for (m, p) = (2,4), (4,2), (2,5), (5,2), (3,3), and for (2,p),
(p,2), (4,p), (p,4) with every even p, there is a non-empty open set of systems that are not stabilizable. Before,
the only pair known to fail was (2,2). The cases (2,4), (2,5), (3,3) rest on explicit integer matrices with
certificates that were checked in exact arithmetic by four programs written independently of each other (finder,
first independent run, lead, second independent run) and by a cross-check with PARI/GP. The two infinite families
rest on proofs written out in the note (dimension counts), supported by exact instances for (2,2), (2,4), (2,6),
(2,8) and (4,2) and by numerical computations for (4,4). All other pairs with d(m,p) even remain open, and the
classification asked for is not obtained. The note is unrefereed. Two AI-assisted independent verification runs
tried to refute every part, found no mathematical error, and asked for changes of presentation, which were made.

## Statement checked
- **Primary source.** A. Eremenko, "Stabilizability by static output feedback", problem file
  https://www.math.purdue.edu/~eremenko/dvi/gpolep.pdf (5 pages, dated October 3, 2026; 135,278 bytes;
  sha256 b03726a3c9d682b9c3ffd99a16d3b395b9278f44eb667e7ac3de20526ce416de; Last-Modified 2026-10-03 10:16:42 GMT).
  It is linked from the list https://www.math.purdue.edu/~eremenko/uns1.html (Last-Modified 2026-10-03 00:12:37
  GMT), section on polynomial matrices and control theory. Both were fetched on 2026-10-03 by the finder, by the
  first independent run, at the writing stage (16:57 GMT) and by the second independent run (19:08 GMT); the copies
  of the problem file are identical. The list does not mark the problem as solved.
  - Setting: real A (n × n), B (n × p), C (m × n), feedback u = Ky with real K (p × m), closed loop A + BKC.
  - Known, according to the file: for n < mp generic pole placement (Wang); for n = mp and even m, p an open set
    of systems where pole placement fails for some pole set (Eremenko–Gabrielov); for m = p = 2 an open set of
    systems that are not stabilizable (Byrnes–Anderson); m = 1 or p = 1 is linear.
  - The question, verbatim: "For which m and p the generic system with n = mp is stabilizable?"
  - Criterion attributed to Byrnes and Anderson: the generic system is stabilizable if and only if for every
    non-degenerate V in Q(m, m+p) the equation det[V(z); L] = c z^{mp} has a real solution L. A system V is called
    degenerate if det[V; L] vanishes identically for some L in G(p, m+p); the file does not say whether L is real
    or complex. The note proves non-degeneracy over C for its three matrices, which is the stronger statement.
  - The file prints a 2 × 4 matrix as the (2,2) counterexample and says that no regular procedure for finding
    such examples is known.
- **Byrnes and Anderson, SIAM J. Control Optim. 22 (1984) 362–380.** Read in full at the writing stage (an
  archived copy of the scan on the second author's publication page); the statements used were compared with the
  scan again in the second independent run. Theorem 1: for mp ≤ n, generic stabilizability is equivalent to the
  existence, for every non-degenerate system, of a gain with closed loop polynomial sⁿ. Theorem 2: mp ≥ n is
  necessary. Theorem 3, attributed to unpublished work of P. Molander: an open set of non-stabilizable systems for
  m = p = 2, n = 4; it is verified in Example 4 by an explicit system (F, G, H). Example 3 leaves (2,4) with n = 8
  open. The paper contains no negative case beyond (2,2).
- **Corpus record.** ulamai/UnsolvedMath, AMR-036-0044 (status `open`). The statement is a faithful paraphrase of
  the question. The summary line of the record says "no (open obstructions) when m, p both even". This is
  inaccurate: for even m, p only the failure of pole placement was known, and non-stabilizability only for (2,2).

## Readings
| Reading | Outcome in the note | Basis |
|---|---|---|
| "generic system stabilizable" = the stabilizable systems are dense (equivalently, the others lie in a proper algebraic set); negation = a non-empty open set of systems that are not stabilizable | used throughout; the same form as the known case (2,2) in the source | Section 1 |
| (m,p) = (2,4), (4,2), (2,5), (5,2), (3,3) | not generically stabilizable | Theorem 1.1, exact certificates |
| (2,p), (p,2), p even | not generically stabilizable | Theorem 1.2, proof |
| (4,p), (p,4), p even | not generically stabilizable | Theorem 1.3, proof |
| all other pairs with d(m,p) even, e.g. (2,9), (3,4), (3,5), (5,5), (6,6) | open | — |
| "stabilizable if and only if d(m,p) is odd" | open; stated as Problem 6.1, not claimed | — |
| the 2 × 4 matrix printed in the problem file (copy dated October 3, 2026) | not a counterexample as printed: it has two real solutions; the example of Byrnes–Anderson corresponds to the matrix with z − 1 in place of z + 1 | Remark 3.3, exact |

## Results in the paper
- **Proposition 2.3** (one direction of the Byrnes–Anderson criterion, full proof). If some m × (m+p) polynomial
  matrix V with entries of degree ≤ p has property (NS), i.e. no real p-plane L gives det[V; L] = c z^{mp} with c
  real, c = 0 included, then there is a non-empty open set of systems with n = mp that are not stabilizable, for
  (m,p) and for (p,m). Neither coprimality of the minors nor non-degeneracy over C is needed.
- **Theorem 1.1.** Explicit integer matrices for (2,4), (2,5), (3,3) with property (NS). The certificate is a
  linear combination Q of Plücker relations that is positive definite on the linear space U(V) (Lemma 3.1). The
  matrices and the multipliers for (2,4) and (2,5) are printed; the 45 forms for (3,3) are defined in the text and
  their multipliers are in the package.
- **Theorems 1.2 and 1.3.** (2,p), (p,2), (4,p), (p,4) for every even p. An a × (a+b) matrix of complex
  polynomials of degree 2b is regarded as a real 2a × (2a+2b) matrix. Lemma 4.1 rewrites the real problem; Lemma
  4.2 states the two inequalities for dimensions of semialgebraic sets that are used (from Hardt's theorem,
  Bochnak–Coste–Roy); Lemma 4.3 and Lemma 5.1 give the codimensions; the counts show that the complex curves with a
  real solution form a set of dimension at most D − 1.
- **Corollary 1.4.** For (2,5), (5,2), (3,3) the real pole placement map is not surjective on a non-empty open set
  of systems. These are cases of Conjecture 2.5 of Rosenthal and Sottile (arXiv:math/9702218).
- **Proposition 4.4 and Example 4.5.** For m = 2 the condition is equivalent to: the space K_v of Hermitian
  matrices contains no non-zero matrix of rank ≤ 2. Explicit examples for (2,2) and (2,4), computed in the text.
- **Remark 5.2.** Exact instances for (4,2); numerical evidence only for (4,4).
- **Problem 6.1** records the question whether generic stabilizability holds exactly when d(m,p) is odd.

## Computations (scripts and outputs in reproducibility/)
Exact arithmetic; all of these were run again on 2026-10-03 from the packaged files.
- **Finder** (`finder/scripts/verify_pd_certificate.py`, pure Python rational arithmetic). For each certificate:
  minors, degrees, gcd; rank M = n+1; dim U(V); Q positive definite on U(V) (all pivots positive); complex
  non-degeneracy by a Macaulay matrix of full column rank modulo a prime. "ALL CHECKS PASSED" for the three matrices
  of Theorem 1.1, for a second certificate each for (2,4) and (2,5), for five further matrices, for the (4,2)
  matrix of Remark 5.2(a) and for the (2,4) matrix of Example 4.5(b); for the last two the complex test is skipped,
  since they are degenerate by construction.

  | file | (m,p) | dim U | smallest pivot of Q on U | Macaulay rank |
  |---|---|---|---|---|
  | cert_24_nice_int | (2,4) | 7 | 0.387 | 126 of 126 (degree 4) |
  | cert_25_nice_int | (2,5) | 11 | 0.501 | 2002 of 2002 (degree 5) |
  | cert_33_nice | (3,3) | 11 | 5.8e-6 | 2002 of 2002 (degree 5) |

  For (3,3) the margin is small (smallest eigenvalue of Q on the unit sphere of U(V) about 1.3e-6). The test is
  exact, so this does not affect the result.
- **PARI/GP cross-check** (`finder/scripts/verify_pari.py`): minors, U(V), leading principal minors, and
  Q(y([I | X])) ≡ 0 as a polynomial identity. All 7 files pass.
- **First independent verification run** (`independent_run/indep_verify.py`, own code): Leibniz minors, own
  elimination, own generator of the Plücker relations, the polynomial identity, positive definiteness by Bareiss
  minors and by LDLᵀ, Macaulay rank modulo another prime. PASS for 8 files.
- **Lead** (`lead/lead_check.py`, standard library): reads the printed matrices and multipliers from `main.tex`
  and repeats the positivity test with its own code (smallest pivots 0.387, 0.501, 5.8e-6); checks the identity
  det(sI − F − GKH) = det[V′(s); K I₂] for the system of Byrnes and Anderson, the conditions stated in Remark 3.3,
  the real solution of the printed matrix in Q(√5), Example 4.5(a),(b), the determinant identities of Lemma 4.1
  and of the types j = 1, 2, the parity criterion for d(m,p) with m, p ≤ 40, and the identities of Step 5 of the
  proof of Proposition 2.3 for the (2,4) matrix with ε = 1/3 and 1/7 (W_I = T_ε((V_0)_I), and
  det(sI − A_0 − B_0KC_0) = Σ_I (Wg)_I(s) y(K)_I for random integer K). 31 checks, all PASS; run again on the final
  text.
- **Second independent verification run** (`independent_run_2/`, own code; see the section below).
- **Complex curves** (`finder/scripts/cc_m2_exact.py`, `cc_m2_exact_b4.py`; `independent_run/indep_m2_exact.py`,
  `indep_m2_b4.py`): b = 1: K_v = 0; b = 2: K_v = R·H₀, v H₀ v* = 2z⁸, det H₀ = −20, signature (2,1); b = 3:
  dim K_v = 4, Macaulay rank 56 of 56 in degree 5; b = 4: dim K_v = 9, Macaulay rank 6435 of 6435 in degree 7.
  Both implementations agree. The (2,4) example also has a positivity certificate, which passes the finder's and
  the independent verifiers.
- **Realizations** (`finder/scripts/realization.py`, `independent_run/check_abc.py`): rational systems (A,B,C)
  with 8, 10, 9 states for the three matrices; the identity with det(zI − A − BKC) holds exactly (κ = −15, −2, 378).

Floating point arithmetic; evidence only.
- **(4,4)** (`independent_run/g44_*.py`, run again in both runs): for nine random complex curves, all 2384
  solutions of 15 of the 16 equations on the 15-dimensional variety X were computed; in every run 2384 distinct
  regular solutions (residual ≤ 2.5e-16), and the 16 equations never vanish together (normalized minimum between
  9.7e-6 and 9.1e-5). Two positive controls with planted real solutions are detected (residual below 3e-16).
  Theorem 1.3 has no machine-checked instance for b ≥ 2.
- Sanity checks of the identities and of the codimensions (Jacobian ranks at random points), the construction of
  Proposition 2.3 for (2,2), homotopy continuation for the certified matrices (14, 42, 42 distinct complex
  solutions, none real), and the perturbation experiment of Example 4.5(d) were run again. The random scans and
  random walks by which the examples were found, and the local least squares searches of the first independent
  run, were not run again; their original outputs are included.

## First independent verification run
One AI-assisted independent verification run examined the finder's write-up, scripts and outputs (2026-10-03). It
wrote its own exact verifier and its own numerical tests, re-derived all dimension counts, and tried to refute
each part.

| Item | Outcome |
|---|---|
| Statement fidelity | confirmed; the claim is a partial answer and says so |
| Proposition 2.3 | confirmed, every step checked; numerical end-to-end test for (2,2) |
| Theorem 1.1 | confirmed by exact certificates with independent code |
| Lemmas 4.1, 4.3, 5.1 | confirmed; numerical confirmation of the codimensions |
| Theorems 1.2 and 1.3 | confirmed; no gap found; exact instances re-verified; new exhaustive numerical test for (4,4) |
| The (2,2) matrix in the problem file | confirmed: two real solutions |
| Novelty | no earlier non-stabilizability result beyond (2,2) found; caveats on sources not read |
| Presentation | fixes required |

Required fixes and what was done.
1. Read Byrnes–Anderson (1984) in full, and try Willems–Hesselink (1978) and Byrnes' survey (1989). Done for
   Byrnes–Anderson: it contains no negative case beyond (2,2), the criterion is its Theorem 1, and its Example 4
   settles the misprint (Remark 3.3). The other two were not accessible; this is stated in the note.
2. Cite Conjecture 2.5 of Rosenthal and Sottile and state the consequence for pole placement. Done (Corollary 1.4),
   with the wording of the conjecture checked in the arXiv source and with a caveat on novelty.
3. Lemma 5.1: write out that the row degrees satisfy η_l ≤ d, and the dimension and freeness of the group action.
   Done. Cite a reference for the inequalities on dimensions of fibres. Done: Lemma 4.2, with what is used.
4. (4,4): replace the least squares paragraph by the exhaustive computation, as numerical evidence only, and say
   that there is no machine-checked instance for b ≥ 2. Done (Remark 5.2(b)).
5. Reword the paragraph on the exceptional (2,4) curve. Done (Example 4.5(d)): only what the saved computation
   shows is stated, and the rest follows from openness.
6. State the conventions at the start: V is m × (m+p), p inputs, m outputs; the meaning of "generic" and of its
   negation; (NS) includes c = 0. Done (Sections 1 and 2).
7. Fetch the source list and the problem file again and repeat the arXiv search on the day of writing. Done on
   2026-10-03.
8. Keep the status of the record open and report partial progress. This was the recommendation of the first run;
   the second run recommends the label `partially_solved`, see below.

## Second independent verification run
A second AI-assisted independent verification run examined the note itself (the 13-page version of 2026-10-03,
`main.tex` sha256 d94a9ca8…06dc), the package and the first report. It wrote its own programs
(`reproducibility/independent_run_2/`), re-derived every count, read the statements quoted from the sources, and
re-ran the package from an extracted copy of `source.zip`. It found no mathematical error.

| Item | Outcome |
|---|---|
| Statement fidelity | confirmed against the problem file fetched at 19:08 GMT (unchanged); the note answers the question as posed, partially, and says so |
| Proposition 2.3 | confirmed step by step (compactness, Möbius substitution, drop of degree, observer form, openness, p inputs and m outputs, transposition); consistent with Theorem 1 of Byrnes–Anderson as printed there; Lemma 2.1 and Steps 4, 5 checked exactly (Step 5 for the three matrices, ε = 1/3, 1/7) |
| Theorem 1.1 | confirmed by a fourth exact verifier (`verify_thm11.py`): data of `main.tex` = certificate files = numbers in the PDF; every R_{I,J} and Q vanish identically on the minors of [I \| X]; dim U(V) = 7, 11, 11; Q positive definite on U(V) by elimination (smallest pivots 0.387, 0.501, 5.8e-6) and by the characteristic polynomial |
| Remark 3.2 | confirmed: minors coprime of degree n; Macaulay ranks 126, 2002, 2002 modulo 2³¹ − 1 |
| Lemmas 4.1–4.3, 5.1, Proposition 4.4 | re-derived, no gap; identities and codimensions confirmed numerically (38 checks) |
| Theorems 1.2 and 1.3 | re-derived, no gap: D − 1 (type (i)), D − 2b (type (ii)); D − 1 (j = 0), D − 2b + 1 (j = 1), D − 4b (j = 2). At planted solutions the differential with respect to Ξ has rank n − 1, so the bound D − 1 is attained |
| Random complex curves, exact | (2,2): (NS) for 1996 of 2000, the other four have K_v ≠ 0; (2,4): 300 of 300; (2,6): 10 of 10 (Macaulay rank 56 over Q); (4,2): exact certificates for 9 of 9 curves |
| (4,4) | no certificate as in Lemma 3.1 found for seven random curves; local searches for real solutions of three random curves (2000 starts each) found none, planted solutions of three controls were found from 12, 8, 5 of 500 starts. Floating point; evidence only |
| Corollary 1.4 | logic confirmed; Conjecture 2.5 of Rosenthal–Sottile quoted correctly; no earlier proof for (2,5), (5,2), (3,3) found; the caveat in the note is kept |
| Remark 3.3 | confirmed exactly (identity for the system of Byrnes–Anderson as printed in their Example 4; Gram determinants 3/4 and −5/4; the real solution with t = (√5 − 1)/2) |
| Example 4.5(d) | the three curves of the experiment have (NS) themselves (exact); for the third one the generator of K_v is nearly singular |
| Re-run of the package | `finder/run_exact.sh`, `run_pari.sh`, `independent_run/run_exact.sh`, `run_sanity.sh`, `run_g44.sh`, `lead/lead_check.py`, and the finder's numerical cross-checks: all outputs reproduced, identical up to run times |
| References | all 16 DOIs resolve at Crossref with matching data |
| Novelty | no earlier non-stabilizability result beyond (2,2) found (see below) |

Required fixes of the second run and what was done.
1. Date the remark on the (2,2) matrix (abstract, Remark 3.3): the copy dated October 3, 2026; say that the file is
   kept on the web and may be corrected. Done.
2. Remark 2.4: do not attribute "complex L" to the problem file. Done.
3. Section 3: define the 45 forms of the certificate for (3,3) and name the files. Done.
4. Example 4.5(d): state that the curve of the experiment has (NS) and that the generator of K_v is nearly
   singular. Done.
5. Remark 5.2: record the nine further exact certificates for (4,2) and the unsuccessful search for (4,4). Done.
6. Verification paragraph: record the second run and what was run again. Done (item 6).
7. Scope paragraph: give the time of the last fetch of the problem file. Done.
8. This report: add the second run; describe the proofs as written out in the note and the checks as AI-assisted
   verification runs; recommend the status `partially_solved`. Done.
9. Package: add `reproducibility/independent_run_2/`; run the lead check and the programs of the second run on the
   final text; rebuild the archive and the deposit files. Done.

The note claims Proposition 2.3 and Theorems 1.1–1.3 with Corollary 1.4 only. It makes no claim for
min(m,p) ≥ 6, for (3,4), (2,9), (3,5), (5,5) or the other open pairs, and no claim of an explicit certified
example for (4,2b) with b ≥ 2.

## Relation to the literature, novelty and scope
- **Known before.** Generic stabilizability for n < mp (Wang 1992) and for n = mp with d(m,p) odd
  (Brockett–Byrnes 1981; d(m,p) is odd if and only if min(m,p) = 1, or min(m,p) = 2 and max(m,p) = 2^k − 1,
  Berstein 1976). mp ≥ n is necessary (Byrnes–Anderson). Not generically stabilizable: (2,2) only (Molander;
  Byrnes–Anderson). Failure of generic real pole placement, a different and stronger property: (2,2)
  (Willems–Hesselink 1978), (4,2) (Rosenthal–Sottile 1998), min(m,p) = 2 and max(m,p) even (Eremenko–Gabrielov,
  Linear Algebra Appl. 2002), all even m, p (Eremenko–Gabrielov, SIAM J. Control Optim. 2002, who ask whether
  generic systems with n = mp and even m, p are stabilizable).
- **Why pole placement results do not settle stabilizability.** A system is not stabilizable only if every stable
  polynomial is omitted. The systems of Eremenko–Gabrielov are close to the Wronski system, which is stabilizable.
- **Sources read at the writing stage.** Eremenko's problem file and list; Byrnes–Anderson 1984 (in full);
  Eremenko–Gabrielov SIAM 2002 (preprint version, in full) and Linear Algebra Appl. 2002 (preprint version,
  the first seven of ten pages); Rosenthal–Sottile (arXiv source, the relevant parts); Rosenthal–Willems, "Open problems in the area
  of pole placement" (1999, in full); Eremenko–Gabrielov, "Degrees of real Wronski maps" (arXiv source, the
  statements used); the arXiv source of Sottile's lecture notes "Real solutions to equations from geometry"
  (searched for pole placement: no occurrence).
- **Not accessible.** Willems–Hesselink 1978, Byrnes 1989, Ravi–Rosenthal–Wang 1994 (publisher pages behind
  access controls; no abstract in zbMATH, Crossref, OpenAlex or Semantic Scholar). According to
  Eremenko–Gabrielov, the last paper solves generic stabilizability for complex feedback. Rosenthal–Willems
  (Section 3.4) say, with reference to Byrnes–Anderson and Ravi–Rosenthal–Wang, that systems that are not
  generically pole assignable have an open set of non-stabilizable systems; Byrnes–Anderson prove this for
  n > mp, and the note reads the statement in this sense. The same section says that pole assignability covers
  stabilizability only in part. The second run read that section and agrees with this reading.
- **Searches.** zbMATH, Crossref, OpenAlex and arXiv, and the works citing Byrnes–Anderson 1984,
  Eremenko–Gabrielov 2002 (both papers) and Rosenthal–Sottile 1998, by the finder and by the first independent run
  on 2026-10-03; arXiv again at the writing stage on 2026-10-03 (static output feedback in 2026; output feedback
  with generic stabilizability; pole placement with Grassmannian, Schubert or Wronski). The second run repeated
  the searches on 2026-10-03: the 193 works citing the four papers and "Degrees of real Wronski maps" (OpenAlex),
  arXiv, zbMATH, Crossref and OpenAlex keyword searches. Nothing on generic stabilizability for n = mp beyond
  (2,2). The 2026 arXiv papers found concern the computational complexity of static output feedback
  stabilization. Four web searches in total, one at each stage.
- **Corollary 1.4.** The pairs (2,5), (5,2), (3,3) were not found in the papers above. The literature on real
  Schubert calculus was not searched exhaustively, and the published version of Sottile's book was not seen.
- **Caveat.** The source list is edited frequently, and the problem file was regenerated on the day of this
  verification. This negative search is not a proof of priority.

## Suggested status of the corpus record
`partially_solved`: new negative cases are settled, and the classification of all pairs (m,p) asked for in the
record remains open for infinitely many pairs. The summary line about even m and p should be corrected:
non-stabilizability is now proved for min(m,p) ∈ {2,4} with m, p even and for (2,5), (5,2), (3,3); before, it was
known only for (2,2), and for the other pairs with even m and p only the failure of pole placement is known.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/