Finiteness theorems for positive definite 𝑛-regular quadratic forms

Chan, Wai Kiu; Oh, Byeong-Kweon

doi:10.1090/s0002-9947-03-03262-8

Cited by 26 publications

(14 citation statements)

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“…More generally, a positive definite integral quadratic form f is called n-regular if f represents all quadratic forms of rank n that are represented by the genus of f . It was proved in [2] that there are only finitely many positive definite primitive n-regular forms of rank n + 3 for n ≥ 2. See also [13] for the structure theorem for n-regular forms in higher rank cases.…”

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Regular positive ternary quadratic forms

2011

Acta Arith.

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Regular positive ternary quadratic forms

2011

Acta Arith.

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“…Recall that we are assuming that the norm npLq of a Z-lattice L is Z. Hence the scale spLq of L is Z or 1 2 Z. Lemma 3.1. Let L be a strongly s-regular ternary Z-lattice with m s pLq " 1.…”

Section: Strongly S-regular Ternary Lattices Representing Onementioning

confidence: 99%

“…Let L be a ternary Z-lattice. Assume that the 1 2 Z p -modular component in a Jordan decomposition of L p is nonzero isotropic. Assume that p is a prime dividing 4dL.…”

Section: Non Trivial Strongly S-regular Ternary Latticesmentioning

confidence: 99%

The number of representations of squares by integral ternary quadratic forms (II)

Kim

2017

Journal of Number Theory

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Abstract. Let f be a positive definite ternary quadratic form. We assume that f is non-classic integral, that is, the norm ideal of f is Z. We say f is strongly s-regular if the number of representations of squares of integers by f satisfies the condition in Cooper and Lam's conjecture in [2]. In this article, we prove that there are only finitely many strongly s-regular ternary forms up to equivalence if the minimum of the non zero squares that are represented by the form is fixed. In particular, we show that there are exactly 207 nonclassic integral strongly s-regular ternary forms that represent one (see Tables 1 and 2). This result might be considered as a complete answer to a natural extension of Cooper and Lam's conjecture.

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“…More generally, a positive definite integral quadratic form f is called n-regular if f represents all quadratic forms of rank n that are represented by the genus of f . It was proved in [3] that there are only finitely many positive definite primitive n-regular forms of rank n + 3 for n ≥ 2. See also [14] on the structure theorem of n-regular forms for higher rank cases.…”

Section: Introductionmentioning

confidence: 99%

Representations of Arithmetic Progressions by Positive Definite Quadratic Forms

2011

Int. J. Number Theory

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For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

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Finiteness theorems for positive definite 𝑛-regular quadratic forms

Cited by 26 publications

References 20 publications

Regular positive ternary quadratic forms

Regular positive ternary quadratic forms

The number of representations of squares by integral ternary quadratic forms (II)

Representations of Arithmetic Progressions by Positive Definite Quadratic Forms

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