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- W845393017 abstract "Let $C$ be a cubic form with rational coefficients in $n$ variables, and let $h$ be the $h$-invariant of $C$. Let $L_1, ldots, L_r$ be linear forms with real coefficients such that if $boldsymbol{alpha} in mathbb{R}^r setminus { boldsymbol{0} }$ then $boldsymbol{alpha} cdot mathbf{L}$ is not a rational form. Assume that $h > 16 + 8 r$. Let $boldsymbol{tau} in mathbb{R}^r$, and let $eta$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $mathbf{x} in [-P,P]^n$ to the system $C(mathbf{x}) = 0, : |mathbf{L}(mathbf{x}) - boldsymbol{tau}| < eta$. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the $h$-invariant condition with the hypothesis $n > 16 + 9 r$, and show that the system has an integer solution. Finally, we show that the values of $mathbf{L}$ at integer zeros of $C$ are equidistributed modulo one in $mathbb{R}^r$, requiring only that $h > 16$." @default.
- W845393017 created "2016-06-24" @default.
- W845393017 creator A5001727428 @default.
- W845393017 date "2016-03-16" @default.
- W845393017 modified "2023-09-25" @default.
- W845393017 title "Equidistribution of values of linear forms on a cubic hypersurface" @default.
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- W845393017 doi "https://doi.org/10.2140/ant.2016.10.421" @default.
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