Algebra & Number Theory
- Algebra Number Theory
- Volume 10, Number 2 (2016), 421-450.
Equidistribution of values of linear forms on a cubic hypersurface
Let be a cubic form with integer coefficients in variables, and let be the -invariant of . Let be linear forms with real coefficients such that, if , then is not a rational form. Assume that . Let , and let be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions to the system , . If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the -invariant condition with the hypothesis and show that the system has an integer solution. Finally, we show that the values of at integer zeros of are equidistributed modulo in , requiring only that .
Algebra Number Theory, Volume 10, Number 2 (2016), 421-450.
Received: 29 April 2015
Revised: 9 November 2015
Accepted: 27 December 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11D25: Cubic and quartic equations
Secondary: 11D75: Diophantine inequalities [See also 11J25] 11J13: Simultaneous homogeneous approximation, linear forms 11J71: Distribution modulo one [See also 11K06] 11P55: Applications of the Hardy-Littlewood method [See also 11D85]
Chow, Sam. Equidistribution of values of linear forms on a cubic hypersurface. Algebra Number Theory 10 (2016), no. 2, 421--450. doi:10.2140/ant.2016.10.421. https://projecteuclid.org/euclid.ant/1510842483