Hilbert's fourteenth problem and finite generation ideals
Bibliographic record
Abstract
Hilbert's fourteenth problem asks whether invariant rings under algebraic group actions are always finitely generated. There are a number of examples that have been constructed since the mid-20th century which demonstrate that this is not the case in general. This thesis is concerned with developing our understanding of these non-finitely generated invariant rings. This goal is ambitious, as by their nature these rings are difficult to work with and it is hard to build an intuition for what might be true in general. The difficulty of trying to develop a solid intuition from examples is exacerbated by the process of ``removing symmetries,'' which relates some of the more well-understood invariant rings. A key construction we employ in order to better understand the structure of these counterexamples to Hilbert's problem is the finite generation ideal, consisting of invariants which make the invariant ring finitely generated after localisation. We take a number of paths in order to achieve our aim, including computing the finite generation ideal for existing examples, constructing new counterexamples, and improving our understanding of both the process of removing symmetries and the finite generation ideal itself. Specifically, we first compute the finite generation ideal of a famous counterexample due to Daigle and Freudenburg. Next, we work on constructing new non-finitely generated invariant rings, focusing primarily on an example proposed by Maubach. We then investigate this process of removing symmetries on some new examples. Finally, we study the finite generation ideal in the setting of monomial algebras, with the intention of passing results obtained to SAGBI-bases; a form of generating set we employ to compute the finite generation ideal for invariant rings.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".