Canada. 1 Positivity in Power Series Rings.
Bibliographic record
Abstract
• Let R[X]: = R[X1, · · · , Xn] be the ring of polynomials in n variables and real coefficients. • A subset M ⊆ R[X] is a quadratic module if 1 ∈ M, M is closed under addition and multiplication by squares (i.e. a 2 f ∈ M, ∀a ∈ R[X] and f ∈ M). • A quadratic preordering is a quadratic module which is also closed under multiplication. • The smallest preordering of R[X] is the set of sums of squares of R[X], denoted by ∑ R[X] 2. 2 • Given a finite subset S = {f1,..., fs} of R[X], the smallest preordering containing S (preordering finitely generated by S) is: TS = { e∈{0,1} s σef e: σe ∈ ∑ R[X] 2, f1, · · · , fs ∈ S} where f e: = f e1 1 · · · f es r, if e = (e1, · · · , es). • Let S = {f1, · · · , fs} ⊂ R[X], S defines a basic closed semialgebraic subset of R n: K = KS = {x ∈ R n: f1(x) ≥ 0,..., fs(x) ≥ 0} 3 • Consider polynomials positive semi-definite on KS: Psd(KS): = {f ∈ R[X] : f(x) ≥ 0 for all x ∈ KS} • Psd(KS) is a preordering in R[X] and TS ⊆ Psd(KS). Hilbert’s 17th Problem is concerned with the issue of representation of positive semi-definite polynomials; motivated by the question: when it true that Psd(KS) = TS? We say: • TS is saturated if Psd(KS) = TS.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".