Counting the number of integer points of non-parametric and parametric polytopes with the PolyhedralSets in Maple
Bibliographic record
Abstract
When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints this task may simply become an undecidable problem. However, studying the integer solutions of systems of equations and inequalities is of practical importance in areas like combinatorial optimization and compiler construction. Over the past 10 years, a series of projects has equipped the computer algebra system Maple with a number of tools for dealing with the integer points of systems of linear equations and inequalities, even in the presence of parameters. With these tools, one can either decide whether integer point solutions exist, or count them, or describe them in a compact way, or enumerate them. In this paper, we will give a tour of these facilities and illustrate their usage with a number of applications. In this paper, we have discussed the adaptation of Barvinok’s algorithm for integer point counting from non-parametric polytopes to parametric polytopes. We report on a Maple implementation of this adaptation. This includes a general framework called Value- UnderConstraints in support of parametric system solving. Additionally, in order to represent the parametric results of these counting of para- metric system, a new Maple data structure called Quasi-polynomial was built.
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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 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 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".