On primitively universal quadratic forms

Budarina, Natalia

doi:10.1007/s10986-010-9076-2

Cited by 8 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The results in this paper will form the basis for many of the arguments in a forthcoming paper in which the authors will give explicit criteria for local primitive universality of an integral quadratic form of any rank. The results obtained there will be further applied in that paper to complete the work initiated by Budarina [3] of determining which among the universal positive definite classically integral quaternary quadratic forms are almost primitively universal. 1.2.…”

mentioning

confidence: 86%

“…In the present paper, we will extend this line of inquiry, revisiting and generalizing the local investigations of Budarina. In the process, we produce new, more elementary proofs of the main results in the paper [3], and extend the results there to the case of forms of even discriminant. In contrast to Budarina's arguments, which draw heavily on Zhuravlev's extensive general work on minimal indecomposable representations, as described in [15] and the references therein, our methods make use of nothing more advanced than standard local theory as presented, for example, in the foundational books of O'Meara [10] and Gerstein [7].…”

mentioning

confidence: 95%

“…As noted above, the identification of almost primitively universal forms reduces completely to a local problem; that is, a positive definite integral quadratic form is almost primitively universal if and only if it is everywhere locally primitively universal. In the paper [3], Budarina began a detailed investigation of the local problem of determining when a form over Z p is primitively universal, and used these results to derive some criteria for a form of odd discriminant to be almost primitively universal. For example, in the spirit of the celebrated fifteen theorem of Conway and Schneeberger (see [5], [1]), Budarina proved: a classically integral quadratic form in at least four variables with odd squarefree discriminant is almost primitively universal if and only if it primitively represents the integers 1, 4 and 8.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On locally primitively universal quadratic forms

Earnest

Gunawardana

2020

Preprint

View full text Add to dashboard Cite

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. The two main results of this paper are: 1) every primitively universal form nontrivially represents zero over every ring Zp of p-adic integers, and 2) every almost universal form in five or more variables is almost primitively universal.

show abstract

mentioning

confidence: 86%

mentioning

confidence: 95%

mentioning

confidence: 99%

See 1 more Smart Citation

On locally primitively universal quadratic forms

Earnest

Gunawardana

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 1 (see [18]). Let Q be a positive definite, classically integral quadratic form of dimension at least 4 with square-free odd determinant.…”

Section: §1 Introductionmentioning

confidence: 99%

“…In our paper [18], we presented an efficient method for determining when a positive definite, classically integral quadratic form of dimension at least four and with square-free odd determinant is primitively almost universal. A quadratic form is said to be (primitively) almost universal if it represents (primitively) all positive integers except for finitely many of them.…”

Section: §1 Introductionmentioning

confidence: 99%

On primitively 2-universal quadratic forms

Budarina

2012

St. Petersburg Math. J.

View full text Add to dashboard Cite

Abstract. The primitive representations of binary positive definite, classically integral quadratic forms over the local rings Z p are studied. For the global ring, an efficient method is obtained for determining when a quadratic form is primitively 2-universal. §1. IntroductionIn 1993, J. Conway and V. Schneeberger announced the 15-theorem, in which they gave a criterion for determining when a positive definite, classically integral (or with an integer matrix) quadratic form represents all positive integers. More precisely, they proved (unpublished) that any positive definite, classically integral quadratic form that represents the set of nine critical numbers 3, 5, 6, 7, 10, 14, 15} is universal. In 1999, M. Bhargava published the paper [1], in which a simpler proof of the 15-theorem was given. He also showed that there exist exactly 204 universal positive definite, classically integral quaternary quadratic forms.Today we can indicate at least four directions in which the universality problem for quadratic forms is being developed:• the multidimensional analog of universal forms (m-universal forms) (see [2,3]);• universal forms over (totally) real number fields (see [4,8]);• almost universal forms (see [9]-[13]);• "even"/"odd"/"prime" universality (representation of all even/odd/ prime numbers) (see [14,15]).Then, the problem of studying primitively universal forms arises naturally. Primitive representations play an important role in finding the number of representations. In [17], Earnest proved that there exist only finitely many 2-regular primitive, positive definite, integral quaternary quadratic forms. Later, B. Kim, M. Kim, and Oh [3] showed that 2-universal quaternary quadratic forms do not exist. (Consequently, dim Q = 5 is the smallest possible dimension for the 2-universal integral quadratic forms Q.) They found all positive definite, classically integral quadratic forms of dimension 5 that represent all positive definite, classically integral, binary quadratic forms (see Table 9), and they 2010 Mathematics Subject Classification. Primary 11E08.

show abstract