Euclidean quadratic forms and ADC forms II: integral forms

Clark, Pete L.; Jagy, William C.

doi:10.4064/aa164-3-4

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2014

2024

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Cited by 4 publications

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“…We say that a quadratic form f /R is ADC (cf. [Cl12], [CJ14]) if for all x ∈ R, whenever there is v ∈ K n such that f (v) = x, there is w ∈ R n such that f (w) = x. In other words, f is ADC if and only if every element of R that is K-represented by f is moreover R-represented by f .…”

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confidence: 99%

Goni: Primes Represented by Binary Quadratic Forms

Clark¹,

Hicks²,

Parshall³

et al. 2014

Integers

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Let f (t 1 , . . . , tn) be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function D f (X), the number of integers of absolute value up to X represented by f . When f is isotropic or n is at least 3, we show that there is a δ(f ) ∈ Q ∩ (0, 1) such that D f (X) ∼ δ(f )X and call δ(f ) the density of f . We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density 0 within the set of locally represented integers.

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confidence: 99%