On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)

Manjul Bhargava

doi:10.5486/PMD.2022.9433

On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)

Manjul Bhargava

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 2⁷². Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large).

Original language	English (US)
Pages (from-to)	513-531
Number of pages	19
Journal	Publicationes Mathematicae Debrecen
Volume	100
Issue number	3-4
DOIs	https://doi.org/10.5486/PMD.2022.9433
State	Published - 2022

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

Thue equation
binary form
invariant order
monogenic
number field
quartic field
quartic order

Access to Document

10.5486/PMD.2022.9433

Cite this

@article{f5255bdb6deb42e98da1ce069c7e7cf4,

title = "On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)",

abstract = "We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 272. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large).",

keywords = "Thue equation, binary form, invariant order, monogenic, number field, quartic field, quartic order",

author = "Manjul Bhargava",

year = "2022",

doi = "10.5486/PMD.2022.9433",

language = "English (US)",

volume = "100",

pages = "513--531",

journal = "Publicationes Mathematicae Debrecen",

issn = "0033-3883",

publisher = "University of Debrecen, Institute of Mathematics",

number = "3-4",

}

TY - JOUR

T1 - On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)

AU - Bhargava, Manjul

PY - 2022

Y1 - 2022

N2 - We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 272. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large).

AB - We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 272. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large).

KW - Thue equation

KW - binary form

KW - invariant order

KW - monogenic

KW - number field

KW - quartic field

KW - quartic order

UR - http://www.scopus.com/inward/record.url?scp=85137744021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85137744021&partnerID=8YFLogxK

U2 - 10.5486/PMD.2022.9433

DO - 10.5486/PMD.2022.9433

M3 - Article

AN - SCOPUS:85137744021

SN - 0033-3883

VL - 100

SP - 513

EP - 531

JO - Publicationes Mathematicae Debrecen

JF - Publicationes Mathematicae Debrecen

IS - 3-4

ER -

On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this