Vishik equivalence and similarity of quadratic forms over fields of characteristic 2
Bibliographic record
Abstract
An important aspect in the algebraic theory of quadratic forms is the study of equivalence relations based on algebraic-geometric properties of the associated quadrics. A well-known criterion originally proved by Vishik in characteristic zero states that two nonsingular quadratic forms of the same dimension have identical Witt indices over all field extensions if and only if their motives are isomorphic in the category of (integral or mod 2) Chow motives. In characteristic 2, it is meaningful to include singular forms. We therefore define two quadratic forms (including singular ones) of the same dimension to be Vishik-equivalent if they share the same isotropy behavior (in a suitably defined way) over all field extensions. Similar quadratic forms are always Vishik-equivalent, but the converse need not hold. We determine various classes of quadratic forms in characteristic 2 where Vishik equivalence implies similarity and give nonsingular counterexamples in all dimensions 2 n ≥ 8 , and also singular counterexamples in dimension 8. To construct the counterexamples, we use a generalized notion of so-called half-neighbors in characteristic 2.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".