On the arithmetic complexity of computing Gröbner bases of comaximal determinantal ideals
Bibliographic record
Abstract
Let M be an n × n matrix of homogeneous linear forms over a field k . If the ideal I n − 2 ( M ) generated by minors of size n − 1 is Cohen-Macaulay, then the Gulliksen-Negård complex is a free resolution of I n − 2 ( M ) . It has recently been shown that by taking into account the syzygy modules for I n − 2 ( M ) which can be obtained from this complex, one can derive a refined signature-based Gröbner basis algorithm DetGB which avoids reductions to zero when computing a grevlex Gröbner basis for I n − 2 ( M ) . In this paper, we establish sharp complexity bounds on DetGB . To accomplish this, we prove several results on the sizes of reduced grevlex Gröbner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fröberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gröbner basis of I n − 2 ( M ) is bounded from below by n 6 asymptotically. The second, also in the zero-dimensional case, states that the complexity of DetGB is bounded from above by n 2 ω + 3 asymptotically, where 2 ≤ ω ≤ 3 is any complexity exponent for matrix multiplication over k .
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".