Ternary universal integral quadratic forms over real quadratic fields

Chan, Wai-kiu; Kim, Myung–Hwan; Raghavan, S.

doi:10.4099/math1924.22.263

Cited by 32 publications

(34 citation statements)

References 9 publications

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“…, x n ) = 1≤i,j ≤n α ij x i x j ( α ij = α ji ∈ O ) is called universal if f represents all integers in O + . Chan, Kim and Raghavan [1] proved that ternary universal forms over F exist if and only if m = 2, 3, 5 and determined all such forms. The question occurs whether there exist quaternary universal forms over real quadratic fields whose discriminants are greater than 12.…”

Section: Introductionmentioning

confidence: 99%

Quaternary universal forms over $\mathbf{Q}(\sqrt{13})$

Sasaki

2007

Ramanujan J

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Let F = Q( √ m) be a real quadratic field over Q with m a square-free positive rational integer and O be the integer ring in F . A totally positive definite integral n-ary quadratic formis called universal if f represents all totally positive integers in O. Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m = 2, 3, 5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12.In this paper we prove that there are only two quaternary universal forms (up to equivalence) over Q( √ 13). For the proof of universality we apply the theory of quadratic lattices.

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Section: Introductionmentioning

confidence: 99%

Quaternary universal forms over $\mathbf{Q}(\sqrt{13})$

Sasaki

2007

Ramanujan J

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“…In 1941, Maass [20] proved the three square theorem, which states: the quadratic form x 2 1 + x 2 2 + x 2 3 is universal over Q( √ 5). All positive definite ternary universal forms over real quadratic fields were determined in [3]. Further developments on universal forms over totally real number fields were established by B. M. Kim (see [12], [13] and [14]).…”

Section: Introductionmentioning

confidence: 99%

The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields

Kim

Park

2009

Math. Comp.

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Abstract. We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields Q( √ −m) for all m. For each imaginary quadratic field Q( √ −m), we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13. In addition, we determine the minimal rank of universal Hermitian lattices for all imaginary quadratic fields.

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“…While no universal positive binary quadratic forms exist, Maass [8] showed that the sum of three squares is universal over Q ( √ 5). In 1994, Chan, Kim and Raghavan [1] showed that among the real quadratic fields, only the fields Q( √ 2), Q( √ 3), and Q( √ 5) admit universal ternary classic integral quadratic forms; all such forms are listed by the authors.…”

Section: Introductionmentioning

confidence: 99%