Inequalities for <i>f</i> <sup>*</sup>-vectors of lattice polytopes
Bibliographic record
Abstract
Abstract The Ehrhart polynomial ehr P (n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f *-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr P (n) with respect to the binomial coefficient basis $\begin{array}{} \bigl\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{d}\bigr\}, \end{array}$ where d = dim P. Similarly to h/h *-vectors, the f *-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f *-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f *-coefficients increases and the last quarter decreases. Even though f *-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h *-vector, there is a polytope with the same h *-vector whose f *-vector is unimodal.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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".