On two forms in many variables of different degrees
Abstract
In this paper, we investigate integral solutions satisfying the system of two forms in many variables of different degrees. Let d1 and d2 be natural numbers with d2>d1≥ 2. Let Fi(x) (i=1,2) be forms in n variables of degrees di (i=1,2), respectively. Define N(F;P):=#{x∈ [−P,P]n∩ ℤn: Fi(x)=0 (i=1,2)}. When each dimension of singular loci of F1=0 and F2=0 is small, we obtain a number n0:=n0(F) such that whenever n> n0 one has the expected asymptotic formula N(F;P)=cF· Pn−d1−d2+O(Pn−d1−d2−δ), for some δ>0, where the constant cF is the product of local densities. We note that this asymptotic formula agrees with the Manin-Peyre conjecture. Compared to the previous work, we lower the admissible threshold n0(F) in most cases, with the exception of case d2−d1=1. In particular, if F1 and F2 are non-singular forms, then we obtain n0(F)=3(d2−1)2d2−1+(d1−1)2d1, provided that d2≥ 5d1 with d1≥2. This yields a substantial improvement over the previous bound n0(F)=(d1+2)(d2−1)2d2−1+d12d1−1. To achieve this, we develop a new differencing argument together with the van der Corput differencing argument, delivering an efficient upper-bound estimate for mean values of exponential sums associated with two forms in many variables of different degrees, when the difference between degrees is sufficiently large. Furthermore, the method described in this paper is flexible enough to apply to forms in many variables of differing degrees in general.
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BibTeX
@misc{yeon2026forms,
title = {{On two forms in many variables of different degrees}},
author = {Kiseok Yeon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17864},
eprint = {2609.17864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17864},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On two forms in many variables of different degrees
AU - Kiseok Yeon
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17864
UR - https://arxiv.org/abs/2609.17864
AB - In this paper, we investigate integral solutions satisfying the system of two forms in many variables of different degrees. Let d_1 and d_2 be natural numbers with d_2>d_1≥ 2. Let F_i(x) (i=1,2) be forms in n variables of degrees d_i (i=1,2), respectively. Define N(F;P):=#{x∈ [−P,P]^n∩ ℤ^n: F_i(x)=0 (i=1,2)}. When each dimension of singular loci of F_1=0 and F_2=0 is small, we obtain a number n_0:=n_0(F) such that whenever n> n_0 one has the expected asymptotic formula N(F;P)=c_F· P^n−d_1−d_2+O(P^n−d_1−d_2−δ), for some δ>0, where the constant c_F is the product of local densities. We note that this asymptotic formula agrees with the Manin-Peyre conjecture. Compared to the previous work, we lower the admissible threshold n_0(F) in most cases, with the exception of case d_2−d_1=1. In particular, if F_1 and F_2 are non-singular forms, then we obtain n_0(F)=3(d_2−1)2^d_2−1+(d_1−1)2^d_1, provided that d_2≥ 5d_1 with d_1≥2. This yields a substantial improvement over the previous bound n_0(F)=(d_1+2)(d_2−1)2^d_2−1+d_12^d_1−1. To achieve this, we develop a new differencing argument together with the van der Corput differencing argument, delivering an efficient upper-bound estimate for mean values of exponential sums associated with two forms in many variables of different degrees, when the difference between degrees is sufficiently large. Furthermore, the method described in this paper is flexible enough to apply to forms in many variables of differing degrees in general.
ER -