[{"intvolume":"       221","OA_type":"green","issue":"2","publisher":"Instytut Matematyczny","keyword":["Diophantine equations","homogeneous forms"],"OA_place":"repository","das_tickbox":"0","project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278","name":"Rational curves via function field analytic number theory"}],"oa":1,"article_type":"original","supplementarymaterial":"no","publication_status":"published","citation":{"ama":"Yamagishi S. Birch’s theorem on forms in many variables with a Hessian condition. <i>Acta Arithmetica</i>. 2025;221(2):141-151. doi:<a href=\"https://doi.org/10.4064/aa241029-19-8\">10.4064/aa241029-19-8</a>","short":"S. Yamagishi, Acta Arithmetica 221 (2025) 141–151.","chicago":"Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian Condition.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2025. <a href=\"https://doi.org/10.4064/aa241029-19-8\">https://doi.org/10.4064/aa241029-19-8</a>.","mla":"Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian Condition.” <i>Acta Arithmetica</i>, vol. 221, no. 2, Instytut Matematyczny, 2025, pp. 141–51, doi:<a href=\"https://doi.org/10.4064/aa241029-19-8\">10.4064/aa241029-19-8</a>.","ista":"Yamagishi S. 2025. Birch’s theorem on forms in many variables with a Hessian condition. Acta Arithmetica. 221(2), 141–151.","ieee":"S. Yamagishi, “Birch’s theorem on forms in many variables with a Hessian condition,” <i>Acta Arithmetica</i>, vol. 221, no. 2. Instytut Matematyczny, pp. 141–151, 2025.","apa":"Yamagishi, S. (2025). Birch’s theorem on forms in many variables with a Hessian condition. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href=\"https://doi.org/10.4064/aa241029-19-8\">https://doi.org/10.4064/aa241029-19-8</a>"},"corr_author":"1","researchdata_availability":"no","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","quality_controlled":"1","year":"2025","doi":"10.4064/aa241029-19-8","page":"141-151","arxiv":1,"date_published":"2025-10-28T00:00:00Z","title":"Birch’s theorem on forms in many variables with a Hessian condition","publication_identifier":{"eissn":["1730-6264"],"issn":["0065-1036"]},"acknowledgement":"The author would like to thank Tim Browning, Jakob Glas and Simon Rydin Myerson for useful suggestions and conversations. Finally, he would like to thank the anonymous referees for their helpful comments. The author was supported by the NWO Veni Grant 016.Veni.192.047 during his time at Utrecht University and by the FWF grant P 36278 at the Institute of Science and Technology Austria while working on this article.","_id":"21768","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2304.02620"}],"oa_version":"Preprint","external_id":{"arxiv":["2304.02620"]},"author":[{"full_name":"Yamagishi, Shuntaro","id":"0c3fbc5c-f7a6-11ec-8d70-9485e75b416b","last_name":"Yamagishi","first_name":"Shuntaro"}],"scopus_import":"1","publication":"Acta Arithmetica","abstract":[{"text":"Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and V∗F the singular locus of the hypersurface {x∈AnC:F(x)=0}. A longstanding result of Birch states that there is a non-trivial integral solution to the equation F(x1,…,xn)=0 provided n>dimV∗F+(d−1)2d, and there is a non-singular solution in R and Qp for all primes p. We give a different formulation of this result. More precisely, we replace dimV∗F with a quantity HF defined in terms of the Hessian matrix of F. This quantity satisfies 0≤HF≤dimV∗F; therefore, we improve on the aforementioned result of Birch if HF<dimV∗F. We also prove the corresponding result for systems of forms of equal degree.","lang":"eng"}],"fulldoi":"https://doi.org/10.4064/aa241029-19-8","date_updated":"2026-07-16T09:04:43Z","day":"28","status":"public","department":[{"_id":"TiBr"}],"language":[{"iso":"eng"}],"month":"10","type":"journal_article","article_processing_charge":"No","volume":221,"date_created":"2026-04-26T22:01:48Z"},{"page":"289-303","year":"2025","doi":"10.4064/aa250115-14-7","quality_controlled":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","researchdata_availability":"no","corr_author":"1","citation":{"ama":"Elsholtz C, Ruzsa IZ, Wurzinger L. Sumset growth in progression-free sets. <i>Acta Arithmetica</i>. 2025;220:289-303. doi:<a href=\"https://doi.org/10.4064/aa250115-14-7\">10.4064/aa250115-14-7</a>","short":"C. Elsholtz, I.Z. Ruzsa, L. Wurzinger, Acta Arithmetica 220 (2025) 289–303.","mla":"Elsholtz, Christian, et al. “Sumset Growth in Progression-Free Sets.” <i>Acta Arithmetica</i>, vol. 220, Institute of Mathematics, 2025, pp. 289–303, doi:<a href=\"https://doi.org/10.4064/aa250115-14-7\">10.4064/aa250115-14-7</a>.","chicago":"Elsholtz, Christian, Imre Z. Ruzsa, and Lena Wurzinger. “Sumset Growth in Progression-Free Sets.” <i>Acta Arithmetica</i>. Institute of Mathematics, 2025. <a href=\"https://doi.org/10.4064/aa250115-14-7\">https://doi.org/10.4064/aa250115-14-7</a>.","apa":"Elsholtz, C., Ruzsa, I. Z., &#38; Wurzinger, L. (2025). Sumset growth in progression-free sets. <i>Acta Arithmetica</i>. Institute of Mathematics. <a href=\"https://doi.org/10.4064/aa250115-14-7\">https://doi.org/10.4064/aa250115-14-7</a>","ieee":"C. Elsholtz, I. Z. Ruzsa, and L. Wurzinger, “Sumset growth in progression-free sets,” <i>Acta Arithmetica</i>, vol. 220. Institute of Mathematics, pp. 289–303, 2025.","ista":"Elsholtz C, Ruzsa IZ, Wurzinger L. 2025. Sumset growth in progression-free sets. Acta Arithmetica. 220, 289–303."},"publication_status":"published","supplementarymaterial":"no","article_type":"original","das_tickbox":"0","isi":1,"publisher":"Institute of Mathematics","OA_type":"closed access","intvolume":"       220","volume":220,"date_created":"2025-11-04T14:33:16Z","article_processing_charge":"No","type":"journal_article","language":[{"iso":"eng"}],"month":"09","status":"public","department":[{"_id":"TiBr"}],"day":"12","date_updated":"2026-07-29T09:51:25Z","fulldoi":"https://doi.org/10.4064/aa250115-14-7","abstract":[{"lang":"eng","text":"We study the growth of sumsets A+B⊂S⊂G, where S does not contain an arithmetic progression of length 2k+1, and where G is a commutative group, in which every nonzero element has an order of at least 2k+1. More specifically, we show the following: if A,B⊂G are sets such that A+B does not contain an arithmetic progression of length 2k+1, then\r\n|A+B|≥|A|2k−13k−2|B|k3k−2.\r\nAs an application we derive upper bounds on the cardinality of the summands in sumsets A+B+C contained in the set of t-th powers, where t≥2 is an integer. In particular, we show that min(|A|,|B|,|C|)≪(logN)4/5 for t=2, and min(|A|,|B|,|C|)≪t(logN)1/2 for t≥3."}],"publication":"Acta Arithmetica","scopus_import":"1","author":[{"first_name":"Christian","last_name":"Elsholtz","full_name":"Elsholtz, Christian"},{"first_name":"Imre Z.","full_name":"Ruzsa, Imre Z.","last_name":"Ruzsa"},{"id":"50c57d72-32a8-11ee-aeea-d652094d2ccd","full_name":"Wurzinger, Lena","first_name":"Lena","last_name":"Wurzinger","orcid":"0009-0004-5360-0074"}],"external_id":{"isi":["001570716800001"]},"oa_version":"None","_id":"20603","acknowledgement":"The authors would like to thank the referee and Ilya Shkredov for comments on the manuscript.\r\nC. E. is supported by a joint FWF-ANR project ArithRand, grant numbers FWF I 4945-N and ANR-20-CE91-0006.\r\n","publication_identifier":{"eissn":["1730-6264"],"issn":["0065-1036"]},"title":"Sumset growth in progression-free sets","date_published":"2025-09-12T00:00:00Z"},{"intvolume":"       207","publisher":"Instytut Matematyczny","keyword":["Algebra","Number Theory"],"isi":1,"oa":1,"article_type":"original","publication_status":"published","citation":{"ieee":"Y. Diao, “Density of the union of positive diagonal binary quadratic forms,” <i>Acta Arithmetica</i>, vol. 207. Instytut Matematyczny, pp. 1–17, 2023.","apa":"Diao, Y. (2023). Density of the union of positive diagonal binary quadratic forms. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href=\"https://doi.org/10.4064/aa210830-24-11\">https://doi.org/10.4064/aa210830-24-11</a>","ista":"Diao Y. 2023. Density of the union of positive diagonal binary quadratic forms. Acta Arithmetica. 207, 1–17.","mla":"Diao, Yijie. “Density of the Union of Positive Diagonal Binary Quadratic Forms.” <i>Acta Arithmetica</i>, vol. 207, Instytut Matematyczny, 2023, pp. 1–17, doi:<a href=\"https://doi.org/10.4064/aa210830-24-11\">10.4064/aa210830-24-11</a>.","chicago":"Diao, Yijie. “Density of the Union of Positive Diagonal Binary Quadratic Forms.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2023. <a href=\"https://doi.org/10.4064/aa210830-24-11\">https://doi.org/10.4064/aa210830-24-11</a>.","short":"Y. Diao, Acta Arithmetica 207 (2023) 1–17.","ama":"Diao Y. Density of the union of positive diagonal binary quadratic forms. <i>Acta Arithmetica</i>. 2023;207:1-17. doi:<a href=\"https://doi.org/10.4064/aa210830-24-11\">10.4064/aa210830-24-11</a>"},"corr_author":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","year":"2023","doi":"10.4064/aa210830-24-11","page":"1-17","date_published":"2023-01-09T00:00:00Z","arxiv":1,"title":"Density of the union of positive diagonal binary quadratic forms","publication_identifier":{"issn":["0065-1036"],"eissn":["1730-6264"]},"acknowledgement":"This article is a version the author’s master thesis at the University of Bonn. The author would like to thank his advisor Valentin Blomer for introducing the problem, and giving generous feedback and encouragement along the way, especially during the global pandemic.\r\nThe author thanks Edgar Assing for his lectures on analytic number theory. Finally, the author is grateful to the anonymous referees for their valuable time and comments.\r\n","_id":"12406","oa_version":"Preprint","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2103.08268","open_access":"1"}],"external_id":{"arxiv":["2103.08268"],"isi":["000912903000001"]},"author":[{"orcid":"0000-0002-4989-5330","last_name":"Diao","first_name":"Yijie","full_name":"Diao, Yijie","id":"7b7eb4ca-eb2c-11ec-b98b-accec0b20c3b"}],"scopus_import":"1","publication":"Acta Arithmetica","abstract":[{"lang":"eng","text":"Let X be a sufficiently large positive integer. We prove that one may choose a subset S of primes with cardinality O(logX) such that a positive proportion of integers less than X can be represented by x2+py2 for at least one p∈S."}],"fulldoi":"https://doi.org/10.4064/aa210830-24-11","day":"09","date_updated":"2024-10-21T06:01:30Z","department":[{"_id":"GradSch"}],"status":"public","language":[{"iso":"eng"}],"month":"01","article_processing_charge":"No","type":"journal_article","date_created":"2023-01-26T21:17:04Z","volume":207},{"oa_version":"Preprint","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2104.14946","open_access":"1"}],"_id":"17058","external_id":{"arxiv":["2104.14946"],"isi":["000844789100001"]},"scopus_import":"1","publication":"Acta Arithmetica","author":[{"orcid":"0000-0002-1812-2810","first_name":"Alec L","last_name":"Shute","id":"440EB050-F248-11E8-B48F-1D18A9856A87","full_name":"Shute, Alec L"}],"date_published":"2022-08-22T00:00:00Z","arxiv":1,"title":"On the leading constant in the Manin-type conjecture for Campana points","publication_identifier":{"eissn":["1730-6264"],"issn":["0065-1036"]},"article_processing_charge":"No","type":"journal_article","date_created":"2024-05-28T13:39:26Z","volume":204,"date_updated":"2026-08-19T12:54:59Z","day":"22","abstract":[{"lang":"eng","text":"We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold (P1,D), where D=1/2[0]+1/2[1]+1/2[∞]. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms."}],"fulldoi":"https://doi.org/10.4064/aa210430-1-7","month":"08","language":[{"iso":"eng"}],"department":[{"_id":"TiBr"}],"status":"public","related_material":{"record":[{"id":"12077","relation":"earlier_version","status":"public"}]},"das_tickbox":"0","oa":1,"isi":1,"article_type":"original","supplementarymaterial":"no","intvolume":"       204","issue":"4","publisher":"Institute of Mathematics","quality_controlled":"1","year":"2022","doi":"10.4064/aa210430-1-7","page":"317-346","researchdata_availability":"no","corr_author":"1","publication_status":"published","citation":{"chicago":"Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for Campana Points.” <i>Acta Arithmetica</i>. Institute of Mathematics, 2022. <a href=\"https://doi.org/10.4064/aa210430-1-7\">https://doi.org/10.4064/aa210430-1-7</a>.","mla":"Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for Campana Points.” <i>Acta Arithmetica</i>, vol. 204, no. 4, Institute of Mathematics, 2022, pp. 317–46, doi:<a href=\"https://doi.org/10.4064/aa210430-1-7\">10.4064/aa210430-1-7</a>.","ista":"Shute AL. 2022. On the leading constant in the Manin-type conjecture for Campana points. Acta Arithmetica. 204(4), 317–346.","ieee":"A. L. Shute, “On the leading constant in the Manin-type conjecture for Campana points,” <i>Acta Arithmetica</i>, vol. 204, no. 4. Institute of Mathematics, pp. 317–346, 2022.","apa":"Shute, A. L. (2022). On the leading constant in the Manin-type conjecture for Campana points. <i>Acta Arithmetica</i>. Institute of Mathematics. <a href=\"https://doi.org/10.4064/aa210430-1-7\">https://doi.org/10.4064/aa210430-1-7</a>","short":"A.L. Shute, Acta Arithmetica 204 (2022) 317–346.","ama":"Shute AL. On the leading constant in the Manin-type conjecture for Campana points. <i>Acta Arithmetica</i>. 2022;204(4):317-346. doi:<a href=\"https://doi.org/10.4064/aa210430-1-7\">10.4064/aa210430-1-7</a>"},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345"},{"_id":"205","oa_version":"None","publication":"Acta Arithmetica","scopus_import":"1","author":[{"first_name":"Timothy D","last_name":"Browning","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177"}],"date_published":"2003-01-01T00:00:00Z","publication_identifier":{"eissn":["1730-6264"],"issn":["0065-1036"]},"title":"Counting rational points on cubic and quartic surfaces","publist_id":"7707","article_processing_charge":"No","type":"journal_article","date_created":"2018-12-11T11:45:12Z","volume":108,"fulldoi":"https://doi.org/10.4064/aa108-3-7","day":"01","date_updated":"2026-05-29T07:22:53Z","status":"public","month":"01","language":[{"iso":"eng"}],"article_type":"original","publisher":"Instytut Matematyczny","OA_type":"closed access","intvolume":"       108","extern":"1","quality_controlled":"1","page":"275 - 295","year":"2003","doi":"10.4064/aa108-3-7","citation":{"ama":"Browning TD. Counting rational points on cubic and quartic surfaces. <i>Acta Arithmetica</i>. 2003;108:275-295. doi:<a href=\"https://doi.org/10.4064/aa108-3-7\">10.4064/aa108-3-7</a>","short":"T.D. Browning, Acta Arithmetica 108 (2003) 275–295.","ista":"Browning TD. 2003. Counting rational points on cubic and quartic surfaces. Acta Arithmetica. 108, 275–295.","ieee":"T. D. Browning, “Counting rational points on cubic and quartic surfaces,” <i>Acta Arithmetica</i>, vol. 108. Instytut Matematyczny, pp. 275–295, 2003.","apa":"Browning, T. D. (2003). Counting rational points on cubic and quartic surfaces. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href=\"https://doi.org/10.4064/aa108-3-7\">https://doi.org/10.4064/aa108-3-7</a>","chicago":"Browning, Timothy D. “Counting Rational Points on Cubic and Quartic Surfaces.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2003. <a href=\"https://doi.org/10.4064/aa108-3-7\">https://doi.org/10.4064/aa108-3-7</a>.","mla":"Browning, Timothy D. “Counting Rational Points on Cubic and Quartic Surfaces.” <i>Acta Arithmetica</i>, vol. 108, Instytut Matematyczny, 2003, pp. 275–95, doi:<a href=\"https://doi.org/10.4064/aa108-3-7\">10.4064/aa108-3-7</a>."},"publication_status":"published","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd"}]