Gröbner-Shirshov bases in some noncommutative algebras
Bibliographic record
Abstract
Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings. This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5). In Chapter 1, we introduce the background and the goals of the thesis. In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp. In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras. In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm. Finally, in Chapter 5, we summarize our results and propose possible future work.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".