There are Salem Numbers with Trace –3 and Every Degree At Least 34

Giacomo Cherubini; Pavlo Yatsyna

doi:10.1080/10586458.2024.2337223

There are Salem Numbers with Trace –3 and Every Degree At Least 34

Giacomo Cherubini^*
, Pavlo Yatsyna

^*Tämän työn vastaava kirjoittaja

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

Abstrakti

We prove that there exist Salem numbers with trace –3 and every even degree (Formula presented.). Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace –3 and degree (Formula presented.), our result is optimal up to possibly the single value 32, for which it is expected there are no such numbers.

Alkuperäiskieli	Englanti
Julkaisu	Experimental Mathematics
DOI - pysyväislinkit	https://doi.org/10.1080/10586458.2024.2337223
Tila	Sähköinen julkaisu (e-pub) ennen painettua julkistusta - 14 huhtik. 2024
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

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10.1080/10586458.2024.2337223Lisenssi: CC BY

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2 Päättynyt

Hollanti NT: Number-theoretic well-rounded lattices
Hollanti, C. (Vastuullinen johtaja), Piispanen, J. (Projektin jäsen), Bolaños, W. (Projektin jäsen), Forst, M. (Projektin jäsen), Costa, M. (Projektin jäsen), Miller, N. (Projektin jäsen), Matala-aho, T. (Projektin jäsen), Koistinen, K. (Projektin jäsen) & Slivinskiy, D. (Projektin jäsen)
01/09/2022 → 31/08/2026
Projekti: RCF Academy Project
MATINE_Hollanti_2022-2023: Kvanttiturvallisten hilasalausmenetelmien analyysi ja verifiointi
Hollanti, C. (Vastuullinen johtaja), Yatsyna, P. (Projektin jäsen), Puniamurthy, K. (Projektin jäsen) & Alpirez Bock, E. (Projektin jäsen)
01/02/2022 → 31/12/2023
Projekti: Other Domestic (20 FC)
Hollanti_ICT: Secure Distributed Computation Schemes with Applications to Digitalized Remote Healthcare
Hollanti, C. (Vastuullinen johtaja), Saarela, P. (Projektin jäsen), Matala-aho, T. (Projektin jäsen), Hieta-aho, E. (Projektin jäsen), Karpuk, D. (Projektin jäsen), Makkonen, O. (Projektin jäsen), Sacikara, E. (Projektin jäsen), Kas Hanna, S. (Projektin jäsen), Allaix, M. (Projektin jäsen), Yatsyna, P. (Projektin jäsen), Niemelä, S. (Projektin jäsen), Rai, M. (Projektin jäsen), Bolaños, W. (Projektin jäsen) & Villamizar Rubiano, D. (Projektin jäsen)
01/01/2021 → 31/12/2023
Projekti: RCF Academy Project

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title = "There are Salem Numbers with Trace –3 and Every Degree At Least 34",

abstract = "We prove that there exist Salem numbers with trace –3 and every even degree (Formula presented.). Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace –3 and degree (Formula presented.), our result is optimal up to possibly the single value 32, for which it is expected there are no such numbers.",

keywords = "cyclotomic, interlacing, polynomial, Salem number, trace",

author = "Giacomo Cherubini and Pavlo Yatsyna",

note = "Publisher Copyright: {\textcopyright} 2024 Taylor \& Francis Group, LLC.",

year = "2024",

month = apr,

day = "14",

doi = "10.1080/10586458.2024.2337223",

language = "English",

journal = "Experimental Mathematics",

issn = "1058-6458",

publisher = "Taylor \& Francis",

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TY - JOUR

T1 - There are Salem Numbers with Trace –3 and Every Degree At Least 34

AU - Cherubini, Giacomo

AU - Yatsyna, Pavlo

PY - 2024/4/14

Y1 - 2024/4/14

N2 - We prove that there exist Salem numbers with trace –3 and every even degree (Formula presented.). Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace –3 and degree (Formula presented.), our result is optimal up to possibly the single value 32, for which it is expected there are no such numbers.

AB - We prove that there exist Salem numbers with trace –3 and every even degree (Formula presented.). Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace –3 and degree (Formula presented.), our result is optimal up to possibly the single value 32, for which it is expected there are no such numbers.

KW - cyclotomic

KW - interlacing

KW - polynomial

KW - Salem number

KW - trace

UR - https://www.scopus.com/pages/publications/85190778117

U2 - 10.1080/10586458.2024.2337223

DO - 10.1080/10586458.2024.2337223

M3 - Article

AN - SCOPUS:85190778117

SN - 1058-6458

JO - Experimental Mathematics

JF - Experimental Mathematics

ER -

There are Salem Numbers with Trace –3 and Every Degree At Least 34

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Projektit

Hollanti NT: Number-theoretic well-rounded lattices

MATINE_Hollanti_2022-2023: Kvanttiturvallisten hilasalausmenetelmien analyysi ja verifiointi

Hollanti_ICT: Secure Distributed Computation Schemes with Applications to Digitalized Remote Healthcare

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