Birch’s theorem on forms in many variables with a Hessian condition

article Birch’s theorem on forms in many variables with a Hessian condition published yes Shuntaro Yamagishi author 0c3fbc5c-f7a6-11ec-8d70-9485e75b416b TiBr department Rational curves via function field analytic number theory project 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. Instytut Matematyczny2025 eng Diophantine equationshomogeneous forms Acta Arithmetica 0065-1036 1730-6264 2304.0262010.4064/aa241029-19-8 2212141-151 Yamagishi S. Birch’s theorem on forms in many variables with a Hessian condition. Acta Arithmetica. 2025;221(2):141-151. doi:<a href="https://doi.org/10.4064/aa241029-19-8">10.4064/aa241029-19-8</a> Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian Condition.” Acta Arithmetica. Instytut Matematyczny, 2025. <a href="https://doi.org/10.4064/aa241029-19-8">https://doi.org/10.4064/aa241029-19-8</a>. Yamagishi S. 2025. Birch’s theorem on forms in many variables with a Hessian condition. Acta Arithmetica. 221(2), 141–151. Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian Condition.” Acta Arithmetica, 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>. Yamagishi, S. (2025). Birch’s theorem on forms in many variables with a Hessian condition. Acta Arithmetica. Instytut Matematyczny. <a href="https://doi.org/10.4064/aa241029-19-8">https://doi.org/10.4064/aa241029-19-8</a> S. Yamagishi, Acta Arithmetica 221 (2025) 141–151. S. Yamagishi, “Birch’s theorem on forms in many variables with a Hessian condition,” Acta Arithmetica, vol. 221, no. 2. Instytut Matematyczny, pp. 141–151, 2025. 217682026-04-26T22:01:48Z2026-07-16T09:04:43Z