[{"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2401.10086"}],"date_created":"2026-06-29T12:55:47Z","year":"2024","publication":"Forum Mathematicum","publication_identifier":{"eissn":["1435-5337"],"issn":["0933-7741"]},"doi":"10.1515/forum-2024-0114","issue":"4","publication_status":"published","external_id":{"arxiv":["2401.10086"]},"abstract":[{"lang":"eng","text":"We consider the set of 𝑚×𝑛 matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions."}],"article_processing_charge":"No","title":"Statistics of ranks, determinants and characteristic polynomials of rational matrices","volume":37,"OA_place":"repository","author":[{"full_name":"Afifurrahman, Muhammad","last_name":"Afifurrahman","first_name":"Muhammad"},{"first_name":"Vivian Zieve","last_name":"Kuperberg","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"},{"last_name":"Ostafe","first_name":"Alina","full_name":"Ostafe, Alina"},{"first_name":"Igor E.","last_name":"Shparlinski","full_name":"Shparlinski, Igor E."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","extern":"1","mathsc":["11C20","15B36","15B52"],"citation":{"apa":"Afifurrahman, M., Kuperberg, V. Z., Ostafe, A., &#38; Shparlinski, I. E. (2024). Statistics of ranks, determinants and characteristic polynomials of rational matrices. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum-2024-0114\">https://doi.org/10.1515/forum-2024-0114</a>","chicago":"Afifurrahman, Muhammad, Vivian Zieve Kuperberg, Alina Ostafe, and Igor E. Shparlinski. “Statistics of Ranks, Determinants and Characteristic Polynomials of Rational Matrices.” <i>Forum Mathematicum</i>. De Gruyter, 2024. <a href=\"https://doi.org/10.1515/forum-2024-0114\">https://doi.org/10.1515/forum-2024-0114</a>.","mla":"Afifurrahman, Muhammad, et al. “Statistics of Ranks, Determinants and Characteristic Polynomials of Rational Matrices.” <i>Forum Mathematicum</i>, vol. 37, no. 4, De Gruyter, 2024, doi:<a href=\"https://doi.org/10.1515/forum-2024-0114\">10.1515/forum-2024-0114</a>.","short":"M. Afifurrahman, V.Z. Kuperberg, A. Ostafe, I.E. Shparlinski, Forum Mathematicum 37 (2024).","ieee":"M. Afifurrahman, V. Z. Kuperberg, A. Ostafe, and I. E. Shparlinski, “Statistics of ranks, determinants and characteristic polynomials of rational matrices,” <i>Forum Mathematicum</i>, vol. 37, no. 4. De Gruyter, 2024.","ama":"Afifurrahman M, Kuperberg VZ, Ostafe A, Shparlinski IE. Statistics of ranks, determinants and characteristic polynomials of rational matrices. <i>Forum Mathematicum</i>. 2024;37(4). doi:<a href=\"https://doi.org/10.1515/forum-2024-0114\">10.1515/forum-2024-0114</a>","ista":"Afifurrahman M, Kuperberg VZ, Ostafe A, Shparlinski IE. 2024. Statistics of ranks, determinants and characteristic polynomials of rational matrices. Forum Mathematicum. 37(4)."},"OA_type":"green","status":"public","oa":1,"date_updated":"2026-07-14T10:48:41Z","publisher":"De Gruyter","arxiv":1,"date_published":"2024-01-01T00:00:00Z","language":[{"iso":"eng"}],"month":"01","type":"journal_article","intvolume":"        37","oa_version":"Preprint","article_type":"original","scopus_import":"1","_id":"22190"},{"publication":"Forum Mathematicum","issue":"3","publication_identifier":{"eissn":["1435-5337"],"issn":["0933-7741"]},"doi":"10.1515/forum-2021-0171","year":"2022","date_created":"2026-06-29T12:57:26Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2107.01437","open_access":"1"}],"quality_controlled":"1","volume":34,"OA_place":"repository","extern":"1","author":[{"full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","last_name":"Kuperberg","first_name":"Vivian Zieve"},{"full_name":"Lalín, Matilde","last_name":"Lalín","first_name":"Matilde"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions","abstract":[{"lang":"eng","text":"In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽𝑞⁢[𝑡] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function 𝑑𝑘⁢(𝑓) over short intervals and over arithmetic progressions for the function field 𝔽𝑞⁢[𝑇] to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function."}],"external_id":{"arxiv":["2107.01437"]},"publication_status":"published","page":"711-747","article_processing_charge":"No","status":"public","day":"26","OA_type":"green","citation":{"mla":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Sums of Divisor Functions and von Mangoldt Convolutions in 𝔽q[T] Leading to Symplectic Distributions.” <i>Forum Mathematicum</i>, vol. 34, no. 3, De Gruyter, 2022, pp. 711–47, doi:<a href=\"https://doi.org/10.1515/forum-2021-0171\">10.1515/forum-2021-0171</a>.","apa":"Kuperberg, V. Z., &#38; Lalín, M. (2022). Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum-2021-0171\">https://doi.org/10.1515/forum-2021-0171</a>","chicago":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Sums of Divisor Functions and von Mangoldt Convolutions in 𝔽q[T] Leading to Symplectic Distributions.” <i>Forum Mathematicum</i>. De Gruyter, 2022. <a href=\"https://doi.org/10.1515/forum-2021-0171\">https://doi.org/10.1515/forum-2021-0171</a>.","ista":"Kuperberg VZ, Lalín M. 2022. Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions. Forum Mathematicum. 34(3), 711–747.","ama":"Kuperberg VZ, Lalín M. Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions. <i>Forum Mathematicum</i>. 2022;34(3):711-747. doi:<a href=\"https://doi.org/10.1515/forum-2021-0171\">10.1515/forum-2021-0171</a>","short":"V.Z. Kuperberg, M. Lalín, Forum Mathematicum 34 (2022) 711–747.","ieee":"V. Z. Kuperberg and M. Lalín, “Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions,” <i>Forum Mathematicum</i>, vol. 34, no. 3. De Gruyter, pp. 711–747, 2022."},"_id":"22194","scopus_import":"1","type":"journal_article","article_type":"original","oa_version":"Preprint","intvolume":"        34","publisher":"De Gruyter","arxiv":1,"date_updated":"2026-07-14T11:02:45Z","month":"03","language":[{"iso":"eng"}],"date_published":"2022-03-26T00:00:00Z","oa":1},{"status":"public","day":"01","citation":{"ista":"Browning TD, Heath-Brown R. 2021. The geometric sieve for quadrics. Forum Mathematicum. 33(1), 147–165.","ama":"Browning TD, Heath-Brown R. The geometric sieve for quadrics. <i>Forum Mathematicum</i>. 2021;33(1):147-165. doi:<a href=\"https://doi.org/10.1515/forum-2020-0074\">10.1515/forum-2020-0074</a>","ieee":"T. D. Browning and R. Heath-Brown, “The geometric sieve for quadrics,” <i>Forum Mathematicum</i>, vol. 33, no. 1. De Gruyter, pp. 147–165, 2021.","short":"T.D. Browning, R. Heath-Brown, Forum Mathematicum 33 (2021) 147–165.","mla":"Browning, Timothy D., and Roger Heath-Brown. “The Geometric Sieve for Quadrics.” <i>Forum Mathematicum</i>, vol. 33, no. 1, De Gruyter, 2021, pp. 147–65, doi:<a href=\"https://doi.org/10.1515/forum-2020-0074\">10.1515/forum-2020-0074</a>.","chicago":"Browning, Timothy D, and Roger Heath-Brown. “The Geometric Sieve for Quadrics.” <i>Forum Mathematicum</i>. De Gruyter, 2021. <a href=\"https://doi.org/10.1515/forum-2020-0074\">https://doi.org/10.1515/forum-2020-0074</a>.","apa":"Browning, T. D., &#38; Heath-Brown, R. (2021). The geometric sieve for quadrics. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum-2020-0074\">https://doi.org/10.1515/forum-2020-0074</a>"},"project":[{"_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","grant_number":"P32428","call_identifier":"FWF","name":"New frontiers of the Manin conjecture"}],"article_type":"original","intvolume":"        33","oa_version":"Preprint","type":"journal_article","_id":"8742","scopus_import":"1","isi":1,"oa":1,"month":"01","language":[{"iso":"eng"}],"date_published":"2021-01-01T00:00:00Z","department":[{"_id":"TiBr"}],"arxiv":1,"publisher":"De Gruyter","date_updated":"2025-04-15T07:39:01Z","year":"2021","date_created":"2020-11-08T23:01:25Z","issue":"1","publication_identifier":{"issn":["0933-7741"],"eissn":["1435-5337"]},"doi":"10.1515/forum-2020-0074","publication":"Forum Mathematicum","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/2003.09593","open_access":"1"}],"author":[{"first_name":"Timothy D","last_name":"Browning","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Heath-Brown","first_name":"Roger","full_name":"Heath-Brown, Roger"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":33,"page":"147-165","article_processing_charge":"No","abstract":[{"lang":"eng","text":"We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros."}],"publication_status":"published","external_id":{"isi":["000604750900008"],"arxiv":["2003.09593"]},"title":"The geometric sieve for quadrics"},{"OA_place":"repository","volume":27,"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","author":[{"orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","first_name":"Timothy D","last_name":"Browning"},{"first_name":"Ritabrata","last_name":"Munshi","full_name":"Munshi, Ritabrata"}],"extern":"1","publication_status":"published","external_id":{"arxiv":["1302.2434"]},"abstract":[{"text":"For suitable pairs of diagonal quadratic forms in eight variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares.","lang":"eng"}],"article_processing_charge":"No","page":"2025 - 2050","title":"Pairs of diagonal quadratic forms and linear correlations among sums of two squares","date_created":"2018-12-11T11:45:28Z","year":"2015","publication":"Forum Mathematicum","publication_identifier":{"issn":["0933-7741"],"eissn":["1435-5337"]},"doi":"10.1515/forum-2013-6024","issue":"4","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1302.2434"}],"type":"journal_article","acknowledgement":"While working on this paper the first author was supported by ERC grant 306457 and the second author was supported by SwarnaJayanti Fellowship 2011–12, DST, Government of India.","oa_version":"Preprint","intvolume":"        27","article_type":"original","scopus_import":"1","_id":"257","publist_id":"7645","oa":1,"date_updated":"2026-05-19T13:08:06Z","publisher":"De Gruyter","arxiv":1,"date_published":"2015-07-10T00:00:00Z","month":"07","language":[{"iso":"eng"}],"OA_type":"green","status":"public","day":"10","citation":{"chicago":"Browning, Timothy D, and Ritabrata Munshi. “Pairs of Diagonal Quadratic Forms and Linear Correlations among Sums of Two Squares.” <i>Forum Mathematicum</i>. De Gruyter, 2015. <a href=\"https://doi.org/10.1515/forum-2013-6024\">https://doi.org/10.1515/forum-2013-6024</a>.","apa":"Browning, T. D., &#38; Munshi, R. (2015). Pairs of diagonal quadratic forms and linear correlations among sums of two squares. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum-2013-6024\">https://doi.org/10.1515/forum-2013-6024</a>","mla":"Browning, Timothy D., and Ritabrata Munshi. “Pairs of Diagonal Quadratic Forms and Linear Correlations among Sums of Two Squares.” <i>Forum Mathematicum</i>, vol. 27, no. 4, De Gruyter, 2015, pp. 2025–50, doi:<a href=\"https://doi.org/10.1515/forum-2013-6024\">10.1515/forum-2013-6024</a>.","short":"T.D. Browning, R. Munshi, Forum Mathematicum 27 (2015) 2025–2050.","ieee":"T. D. Browning and R. Munshi, “Pairs of diagonal quadratic forms and linear correlations among sums of two squares,” <i>Forum Mathematicum</i>, vol. 27, no. 4. De Gruyter, pp. 2025–2050, 2015.","ama":"Browning TD, Munshi R. Pairs of diagonal quadratic forms and linear correlations among sums of two squares. <i>Forum Mathematicum</i>. 2015;27(4):2025-2050. doi:<a href=\"https://doi.org/10.1515/forum-2013-6024\">10.1515/forum-2013-6024</a>","ista":"Browning TD, Munshi R. 2015. Pairs of diagonal quadratic forms and linear correlations among sums of two squares. Forum Mathematicum. 27(4), 2025–2050."}},{"page":"881-919","article_processing_charge":"No","abstract":[{"lang":"eng","text":"We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|^4/d u to blow up. If m0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in  is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in  for the defocusing NLS in three and higher dimensions with spherically symmetric data."}],"publication_status":"published","external_id":{"arxiv":["math/0609690"]},"title":"Minimal-mass blowup solutions of the mass-critical NLS","extern":"1","author":[{"full_name":"Tao, Terence","last_name":"Tao","first_name":"Terence"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","first_name":"Monica"},{"full_name":"Zhang, Xiaoyi","last_name":"Zhang","first_name":"Xiaoyi"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":20,"OA_place":"repository","mathsc":["35Q55"],"quality_controlled":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.math/0609690","open_access":"1"}],"year":"2008","date_created":"2026-06-19T07:53:12Z","issue":"5","publication_identifier":{"eissn":["1435-5337"],"issn":["0933-7741"]},"doi":"10.1515/forum.2008.042","publication":"Forum Mathematicum","das_tickbox":"1","oa":1,"language":[{"iso":"eng"}],"month":"11","date_published":"2008-11-03T00:00:00Z","publisher":"De Gruyter","arxiv":1,"date_updated":"2026-06-25T08:15:22Z","article_type":"original","oa_version":"Preprint","intvolume":"        20","type":"journal_article","_id":"22049","scopus_import":"1","citation":{"short":"T. Tao, M. Vişan, X. Zhang, Forum Mathematicum 20 (2008) 881–919.","ieee":"T. Tao, M. Vişan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical NLS,” <i>Forum Mathematicum</i>, vol. 20, no. 5. De Gruyter, pp. 881–919, 2008.","ista":"Tao T, Vişan M, Zhang X. 2008. Minimal-mass blowup solutions of the mass-critical NLS. Forum Mathematicum. 20(5), 881–919.","ama":"Tao T, Vişan M, Zhang X. Minimal-mass blowup solutions of the mass-critical NLS. <i>Forum Mathematicum</i>. 2008;20(5):881-919. doi:<a href=\"https://doi.org/10.1515/forum.2008.042\">10.1515/forum.2008.042</a>","chicago":"Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.” <i>Forum Mathematicum</i>. De Gruyter, 2008. <a href=\"https://doi.org/10.1515/forum.2008.042\">https://doi.org/10.1515/forum.2008.042</a>.","apa":"Tao, T., Vişan, M., &#38; Zhang, X. (2008). Minimal-mass blowup solutions of the mass-critical NLS. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum.2008.042\">https://doi.org/10.1515/forum.2008.042</a>","mla":"Tao, Terence, et al. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.” <i>Forum Mathematicum</i>, vol. 20, no. 5, De Gruyter, 2008, pp. 881–919, doi:<a href=\"https://doi.org/10.1515/forum.2008.042\">10.1515/forum.2008.042</a>."},"OA_type":"green","day":"03","status":"public"}]