Infinite matroids in tropical differential algebra
Bibliographic record
Abstract
Abstract We consider a finite-dimensional vector space $W\subset K^E$ over a field K and a set E . We show that the set $\mathcal {C}(W)\subset 2^E$ of minimal supports of W are the circuits of a matroid on E . When the cardinality of K is large (compared to that of E ), then the family of supports of W is a matroid. Afterwards we apply these results to tropical differential algebraic geometry ( tdag ), studying the set of supports of spaces of formal power series solutions $\text {Sol}(\Sigma )$ of systems of linear differential equations ( lde s) $\Sigma$ in variables $x_1,\ldots ,x_n$ having coefficients in . If $\Sigma $ is of differential type zero, then the set $\mathcal {C}(Sol(\Sigma ))\subset (2^{\mathbb {N}^{m}})^n$ of minimal supports defines a matroid on $E=[n]\times \mathbb {N}^{m}$ , and if the cardinality of K is large enough, then the set of supports is also a matroid on E . By applying the fundamental theorem of tdag ( fttdag ), we give a necessary condition under which the set of solutions $Sol(U)$ of a system U of tropical lde s is a matroid. We give a counterexample to the fttdag for systems $\Sigma $ of lde s over countable fields for which is not a matroid.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".