New lower bounds for the Schur-Siegel-Smyth trace problem
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- by Bryce Joseph Orloski, Naser Talebizadeh Sardari and Alexander Smith;
- Math. Comp. 94 (2025), 2005-2040
- DOI: https://doi.org/10.1090/mcom/4004
- Published electronically: August 6, 2024
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Previous version: Original version posted August 6, 2024
Corrected version: This version corrects an error in author email addresses.
Abstract:
We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known bound to 1.80203. Our method adds new constraints to Smyth’s linear programming method to decrease the number of variables required in the new problem of interest. This allows for faster convergence recovering Schur’s bound in the simplest case and Siegel’s bound in the second simplest case of our new family of bounds. We also prove the existence of a unique optimal solution to our newly phrased problem and express the optimal solution in terms of polynomials. Lastly, we solve this new problem numerically with a gradient descent algorithm to attain the new bound 1.80203.References
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Bibliographic Information
- Bryce Joseph Orloski
- Affiliation: Penn State Department of Mathematics, McAllister Building, Pollock Rd, State College, Pennsylvania 16802
- ORCID: 0000-0003-0228-410X
- Email: bjo5149@psu.edu
- Naser Talebizadeh Sardari
- Affiliation: Penn State Department of Mathematics, McAllister Building, Pollock Rd, State College, Pennsylvania 16802
- MR Author ID: 1260714
- Email: nzt5208@psu.edu
- Alexander Smith
- Affiliation: UCLA Department of Mathematics, 520 Portola Plaza, Los Angeles, California 90095
- MR Author ID: 1086111
- ORCID: 0000-0002-7694-6754
- Email: asmith13@math.ucla.edu
- Received by editor(s): March 7, 2024
- Received by editor(s) in revised form: June 21, 2024
- Published electronically: August 6, 2024
- Additional Notes: This work was partially supported by NSF grant DMS-2401242.
- © Copyright 2024 American Mathematical Society
- Journal: Math. Comp. 94 (2025), 2005-2040
- MSC (2020): Primary 11Y60, 46N10; Secondary 65E05
- DOI: https://doi.org/10.1090/mcom/4004