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Representations of integers by systems of three quadratic forms

2016; Wiley; Volume: 113; Issue: 3 Linguagem: Inglês

10.1112/plms/pdw027

1460-244X

Lillian B. Pierce, Damaris Schindler, Melanie Matchett Wood,

Limits and Structures in Graph Theory

Proceedings of the London Mathematical SocietyVolume 113, Issue 3 p. 289-344 Articles Representations of integers by systems of three quadratic forms Lillian B. Pierce, Corresponding Author Lillian B. Pierce pierce@math.duke.edu Mathematics Department, Duke University, 120 Science Drive, Durham, NC, 27708 USApierce@math.duke.eduSearch for more papers by this authorDamaris Schindler, Damaris Schindler damaris.schindler@hcm.uni-bonn.de Hausdorff Center for Mathematics, Endenicher Allee 60-62, 53115 Bonn, GermanySearch for more papers by this authorMelanie Matchett Wood, Melanie Matchett Wood mmwood@math.wisc.edu Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Dr., Madison, WI, 53706 USA American Institute of Mathematics, 600 East Brokaw Road, San Jose, CA, 95112 USASearch for more papers by this author Lillian B. Pierce, Corresponding Author Lillian B. Pierce pierce@math.duke.edu Mathematics Department, Duke University, 120 Science Drive, Durham, NC, 27708 USApierce@math.duke.eduSearch for more papers by this authorDamaris Schindler, Damaris Schindler damaris.schindler@hcm.uni-bonn.de Hausdorff Center for Mathematics, Endenicher Allee 60-62, 53115 Bonn, GermanySearch for more papers by this authorMelanie Matchett Wood, Melanie Matchett Wood mmwood@math.wisc.edu Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Dr., Madison, WI, 53706 USA American Institute of Mathematics, 600 East Brokaw Road, San Jose, CA, 95112 USASearch for more papers by this author First published: 05 August 2016 https://doi.org/10.1112/plms/pdw027Citations: 1 2010 Mathematics Subject Classification 11D85, 11P55 (primary). Lillian B. Pierce is partially supported by NSF DMS-1402121. Damaris Schindler is partially supported by NSF DMS-1128155. Melanie Matchett Wood is supported by an American Institute of Mathematics Five-Year Fellowship, a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, and National Science Foundation grant DMS-1301690. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers ( n 1 , … , n R ) by a system of quadratic forms Q 1 , … , Q R in k variables, as long as k is sufficiently large with respect to R; reducing the required number of variables remains a significant open problem. In this work, we consider the case of three forms and improve on the classical result by reducing the number of required variables to k ⩾ 10 for ‘almost all’ tuples, under a non-singularity assumption on the forms Q 1 , Q 2 , Q 3 . To accomplish this, we develop a three-dimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms. Citing Literature Volume113, Issue3September 2016Pages 289-344 RelatedInformation