On weighted forms in many variables
Abstract
In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let F(x ; y) be a given weighted form of degree d in variables x ∈ ℝs1 and y ∈ ℝs2, where variables x and y have weights w1 and w2 with w1<w2,(w1, w2)=1, and d>w1 w2. Write RF(P):=#{(x ; y) ∈ ℤs1+s2: F(x ; y)=0,|x| ≤ Pw1 / d,|y| ≤ Pw2 / d} . In particular, we show that whenever s1+s2−σF>(1+w2/w1) d/w1 2d / w1, where σF is the dimension of the affine singular locus of F, the quantity RF(P) has the expected asymptotic formula, that is RF(P)=c Ps1 w1 / d+s2 w2 / d−1+o(Ps1 w1 / d+s2 w2 / d−1), where c is the product of local densities. Furthermore, the constant c is positive whenever F(x ; y)=0 has a nonsingular solution over ℝ and ℚp for every prime p. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by F(x ; y)=0 in a weighted projective space of sufficiently large dimensions.
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BibTeX
@misc{galiote2026weighted,
title = {{On weighted forms in many variables}},
author = {Daniel Flores Galiote and Kiseok Yeon},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28774},
eprint = {2608.28774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28774},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On weighted forms in many variables
AU - Daniel Flores Galiote
AU - Kiseok Yeon
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28774
UR - https://arxiv.org/abs/2608.28774
AB - In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let F(x ; y) be a given weighted form of degree d in variables x ∈ ℝ^s_1 and y ∈ ℝ^s_2, where variables x and y have weights w_1 and w_2 with w_1<w_2,(w_1, w_2)=1, and d>w_1 w_2. Write R_F(P):=#{(x ; y) ∈ ℤ^s_1+s_2: F(x ; y)=0,|x| ≤ P^w_1 / d,|y| ≤ P^w_2 / d} . In particular, we show that whenever s_1+s_2−σ_F>(1+w_2/w_1) d/w_1 2^d / w_1, where σ_F is the dimension of the affine singular locus of F, the quantity R_F(P) has the expected asymptotic formula, that is R_F(P)=c P^s_1 w_1 / d+s_2 w_2 / d−1+o(P^s_1 w_1 / d+s_2 w_2 / d−1), where c is the product of local densities. Furthermore, the constant c is positive whenever F(x ; y)=0 has a nonsingular solution over ℝ and ℚ_p for every prime p. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by F(x ; y)=0 in a weighted projective space of sufficiently large dimensions.
ER -