Least Zero of a Cubic Form

Abstract

An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form C ∈ ℤ[X₁,…,X_n], provided that n ≥ 14.

Keywords

Cubic forms / least non-trivial zero / circle method / 11D72 / 11D25 / 11P55

References

[1] Bernert C., Small solutions of homogeneous and inhomogeneous cubic equations. 2023, arXiv:2310.02039

[2] Browning TD, Dietmann R, Elliott PDTA. Least zero of a cubic form. Math. Ann., 2012, 352(3): 745-778

[3] Davenport H. Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2005, Cambridge, Cambridge University Press

[4] Heath-Brown DR. Cubic forms in 14 variables. Invent. Math., 2007, 170(1): 199-230

[5] Lloyd DP. Bounds for solutions of Diophantine equations, 1976, Adelaide, University of Adelaide

RIGHTS & PERMISSIONS

Peking University