Positive Values of Inhomogeneous Quadratic Forms of Type (1, 4): The Constant Γ_{1,4} = 8
Abstract
Let Q be a real indefinite quadratic form of type (1, 4) with determinant D \neq 0. We prove that for every c \in \mathbb{R}^5 there exists x \in \mathbb{Z}^5 such that 0 < Q(x + c) \le (8|D|)^{1/5}, and that the constant 8 is optimal. This establishes a conjecture of Bambah, Dumir and Hans-Gill and, together with earlier work, completes the determination of the sharp constants Γ_{r,s} for positive inhomogeneous values of nondegenerate indefinite quadratic forms. The proof combines classical reduction and gap theorems with uniform bounds for circular gaps of quadratic residues. A central step identifies the exceptional parameters in certain resonant reduced families: an explicit ladder of integer witnesses resolves their punctured neighbourhoods, leaving only three rational parameter points, subject to stated coefficient conditions. These are treated by retaining the coupling in a preceding, less reduced form. The remaining covering statements are verified by finite certificates in exact rational arithmetic.
Related
- S-squareful values of binary quadratic forms
- On the polynomial values represented by binary quadratic forms
- Binary Quadratic Forms, Shintani Lifts, and Twisted L-Values
- Arithmeticity of critical L-values for Hilbert-Siegel modular forms in the left half plane and s = 1
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
BibTeX
@misc{kahlon2026positive,
title = {{Positive Values of Inhomogeneous Quadratic Forms of Type (1, 4): The Constant Γ_1,4 = 8}},
author = {Vineet Kahlon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41223},
eprint = {2610.41223},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41223},
}
RIS
TY - UNPB
TI - Positive Values of Inhomogeneous Quadratic Forms of Type (1, 4): The Constant Γ_{1,4} = 8
AU - Vineet Kahlon
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41223
UR - https://arxiv.org/abs/2610.41223
AB - Let Q be a real indefinite quadratic form of type (1, 4) with determinant D \neq 0. We prove that for every c \in \mathbb{R}^5 there exists x \in \mathbb{Z}^5 such that 0 < Q(x + c) \le (8|D|)^{1/5}, and that the constant 8 is optimal. This establishes a conjecture of Bambah, Dumir and Hans-Gill and, together with earlier work, completes the determination of the sharp constants Γ_{r,s} for positive inhomogeneous values of nondegenerate indefinite quadratic forms. The proof combines classical reduction and gap theorems with uniform bounds for circular gaps of quadratic residues. A central step identifies the exceptional parameters in certain resonant reduced families: an explicit ladder of integer witnesses resolves their punctured neighbourhoods, leaving only three rational parameter points, subject to stated coefficient conditions. These are treated by retaining the coupling in a preceding, less reduced form. The remaining covering statements are verified by finite certificates in exact rational arithmetic.
ER -