[challenge]: A Positivstellensatz beyond Archimedean for state polynomials

Released by

Jie Wang (AMSS, Chinese Academy of Sciences) & Jin-Guo Liu (Hong Kong University of Science and Technology (Guangzhou))

Contact email

cacate0129@gmail.com

Method

Other

Challenge issue

Difficulty: ★★ (rated by Jie Wang)

Background

Krivine–Stengle-type denominator certificates provably fail for state polynomials (arXiv:2301.12513), and the unbounded tracial setting needs bespoke machinery (arXiv:2205.05959). The classical commutative trichotomy (Artin, Krivine–Stengle, Putinar) has no state-polynomial analogue — the deepest purely mathematical hole in the area (survey: arXiv:2412.12342).

Research objective

Find the correct general positivity certificate for state/trace polynomials without Archimedean assumptions. Experimental-mathematics loop: enumerate low-degree globally positive state polynomials, attempt certificates in candidate formats (weighted SOHS + commutators + state-product denominators), and let systematic failures shape the conjectured format — the path that historically led to Artin's theorem — then attack the revealed structure by hand.

Verification plan

• Success gate: a proved Positivstellensatz whose certificate, for every corpus instance, is found by the associated SDP and verifies in exact arithmetic — theorem plus machine-checked instance coverage; any positive corpus polynomial with no certificate in the format falsifies it.
• Hope signal: a format covering strictly more of the corpus than all known formats, with separating examples characterized.
• Pivot signal: systematic counterexamples to every finitary candidate at degree 4 already — pivot to proving the impossibility theorem.

Why this may lead to research output

Whoever finds the right certificate format defines the next decade of duality theory for expectation-valued optimization. Beautiful mathematics with a built-in experimental pipeline.

References

1. Klep, Magron, Volčič, Wang, State polynomials, arXiv:2301.12513 (Math. Prog.).
2. Klep, Scheiderer, Volčič, Globally trace-positive noncommutative polynomials, arXiv:2205.05959.
3. Huber et al., Positivity of state, trace, and moment polynomials, arXiv:2412.12342.

No files, tests, or entry points are named. Start with the cited papers and the low-degree corpus/SDP experiment described in the objective; done means a proved certificate theorem with exact-arithmetic verification for every corpus instance, or a proved impossibility result.