[{"arxiv":1,"year":"2025","oa":1,"date_published":"2025-10-28T00:00:00Z","volume":221,"publication_status":"published","article_type":"original","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"}],"external_id":{"arxiv":["2304.02620"]},"oa_version":"Preprint","researchdata_availability":"no","OA_place":"repository","day":"28","month":"10","date_created":"2026-04-26T22:01:48Z","author":[{"first_name":"Shuntaro","last_name":"Yamagishi","full_name":"Yamagishi, Shuntaro","id":"0c3fbc5c-f7a6-11ec-8d70-9485e75b416b"}],"doi":"10.4064/aa241029-19-8","project":[{"grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory"}],"type":"journal_article","intvolume":"       221","_id":"21768","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2304.02620","open_access":"1"}],"publisher":"Instytut Matematyczny","das_tickbox":"0","keyword":["Diophantine equations","homogeneous forms"],"status":"public","publication":"Acta Arithmetica","page":"141-151","citation":{"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.","short":"S. Yamagishi, Acta Arithmetica 221 (2025) 141–151.","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>","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>","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."},"scopus_import":"1","issue":"2","corr_author":"1","department":[{"_id":"TiBr"}],"title":"Birch’s theorem on forms in many variables with a Hessian condition","publication_identifier":{"issn":["0065-1036"],"eissn":["1730-6264"]},"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.","date_updated":"2026-07-16T09:04:43Z","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"quality_controlled":"1","supplementarymaterial":"no","OA_type":"green"}]