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The Polynomials of Mahler and Roots of Unity

2015· article· en· W2402211513 on OpenAlex
Karl Dilcher, Larry Ericksen

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

aboutThe title or abstract carries a Canadian signal from the geographic lexicon.
no affNo Canadian affiliation: this work is invisible to an affiliation-only frame.
No Canadian affiliation. An affiliation-only frame, the usual design, would never have seen this work. It is one of the works that make the case for inverting the frame.

Bibliographic record

VenueAmerican Mathematical Monthly · 2015
Typearticle
Languageen
FieldMathematics
TopicAlgebraic and Geometric Analysis
Canadian institutionsnot available
Fundersnot available
KeywordsRoot of unityMathematicsPure mathematicsPhysics

Abstract

fetched live from OpenAlex

AbstractKurt Mahler in 1982 studied a special sequence of very sparse (0,1)-polynomials which have only powers of 2 as exponents. In this paper we study divisibility properties of these polynomials by certain cyclotomic polynomials and prove an explicit version of a result that was given only implicitly by Mahler. We also consider the distribution of real and complex noncyclotomic zeros, improving some of Mahler’s results. Then we show that the derivatives of the polynomials of Mahler have all their zeros inside the unit circle. We conclude this paper with some further remarks and open questions. Additional informationNotes on contributorsKarl DilcherKARL DILCHER received his undergraduate education at the Technische Universität Clausthal in Germany. He then did his graduate studies at Queen’s University in Kingston, Ontario, and finished his Ph.D. there in 1983 under the supervision of Paulo Ribenboim. He is currently a professor at Dalhousie University in Halifax, Nova Scotia, Canada, where he first arrived in 1984 as a postdoctoral fellow. His research interests include classical analysis, special functions, and elementary and computational number theory.Larry EricksenLARRY ERICKSEN received his BS degree in Engineering Physics at Cornell University, where he helped construct the university’s high energy particle accelerator. Later he obtained an MBA degree in International Business Development from the Wharton School at the University of Pennsylvania. He joined the private sector as a management consultant and held executive positions in industry, both domestically and overseas. Now in his retirement, he serves on the boards of environmental organizations, art museums, and multicultural institutions.

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.

Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.004
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.051
Threshold uncertainty score0.457

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.004
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0010.000
Bibliometrics0.0000.001
Science and technology studies0.0000.001
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.045
GPT teacher head0.312
Teacher spread0.267 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it