HOMOGENEOUS FORM INEQUALITIES (I) : VOLUME ESTIMATES AND THE GENERALISED METRIC OPPENHEIM CONJECTURE
(2025) In Transactions of the American Mathematical Society 378(4). p.2643-2693- Abstract
This is the first part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, sharp estimates for the volume of the semialgebraic domain under consideration are established. These estimates are expressed as a function of the roots of a Sato–Bernstein polynomial attached to the homogeneous forms. This is then employed to derive, following known methods relying on moment estimates in the space of unimodular lattices, metric statements dealing both with the existence and the number of solutions in integer vectors to the system of inequalities. (In the... (More)
This is the first part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, sharp estimates for the volume of the semialgebraic domain under consideration are established. These estimates are expressed as a function of the roots of a Sato–Bernstein polynomial attached to the homogeneous forms. This is then employed to derive, following known methods relying on moment estimates in the space of unimodular lattices, metric statements dealing both with the existence and the number of solutions in integer vectors to the system of inequalities. (In the forthcoming second part of the work, a deterministic study of the same questions is undertaken.) The results thus established solve problems posed by Athreya and Margulis (2018) on the existence of effective and uniform generalisations of the metric Oppenheim Conjecture.
(Less)
- author
- Adiceam, Faustin and Marmon, Oscar LU
- organization
- publishing date
- 2025-04
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Transactions of the American Mathematical Society
- volume
- 378
- issue
- 4
- pages
- 51 pages
- publisher
- American Mathematical Society (AMS)
- external identifiers
-
- scopus:105000424936
- ISSN
- 0002-9947
- DOI
- 10.1090/tran/9359
- language
- English
- LU publication?
- yes
- id
- af8cee62-c4ce-4669-8edd-9f6c5707328a
- date added to LUP
- 2026-01-09 12:00:59
- date last changed
- 2026-01-09 12:02:06
@article{af8cee62-c4ce-4669-8edd-9f6c5707328a,
abstract = {{<p>This is the first part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, sharp estimates for the volume of the semialgebraic domain under consideration are established. These estimates are expressed as a function of the roots of a Sato–Bernstein polynomial attached to the homogeneous forms. This is then employed to derive, following known methods relying on moment estimates in the space of unimodular lattices, metric statements dealing both with the existence and the number of solutions in integer vectors to the system of inequalities. (In the forthcoming second part of the work, a deterministic study of the same questions is undertaken.) The results thus established solve problems posed by Athreya and Margulis (2018) on the existence of effective and uniform generalisations of the metric Oppenheim Conjecture.</p>}},
author = {{Adiceam, Faustin and Marmon, Oscar}},
issn = {{0002-9947}},
language = {{eng}},
number = {{4}},
pages = {{2643--2693}},
publisher = {{American Mathematical Society (AMS)}},
series = {{Transactions of the American Mathematical Society}},
title = {{HOMOGENEOUS FORM INEQUALITIES (I) : VOLUME ESTIMATES AND THE GENERALISED METRIC OPPENHEIM CONJECTURE}},
url = {{http://dx.doi.org/10.1090/tran/9359}},
doi = {{10.1090/tran/9359}},
volume = {{378}},
year = {{2025}},
}