On the central exponent of superalgebras with superinvolution
Bibliographic record
Abstract
Abstract The growth of central polynomials for matrix algebras over a field of characteristic zero was first studied by Regev in 2016 $2016$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>2016</mml:mn> </mml:math> . This problem can be generalized by analyzing the behavior of the dimension c Subscript n Superscript z Baseline left parenthesis upper A right parenthesis $c_n^z(A)$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> <mml:mi>z</mml:mi> </mml:msubsup> <mml:mo stretchy="false" form="prefix" fence="true">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false" form="postfix" fence="true">)</mml:mo> </mml:mrow> </mml:math> of the space of multilinear polynomials of degree n modulo the central polynomials of an algebra A . In 2018 $2018$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>2018</mml:mn> </mml:math> , Giambruno and Zaicev established the existence of the limit limit Underscript n right arrow infinity Endscripts RootIndex n StartRoot c Subscript n Superscript z Baseline left parenthesis upper A right parenthesis EndRoot period $\lim \limits _{n \to \infty }\sqrt [n]{c_n^{z}(A)}.$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:munder> <mml:mi>lim</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mo>∞</mml:mo> </mml:mrow> </mml:munder> <mml:mroot> <mml:mrow> <mml:msubsup> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> <mml:mi>z</mml:mi> </mml:msubsup> <mml:mo stretchy="false" form="prefix" fence="true">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false" form="postfix" fence="true">)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mroot> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In this article, we extend this framework to superalgebras equipped with a superinvolution, proving both the existence and the finiteness of the corresponding limit.
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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 | 0.000 | 0.001 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.006 | 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".