scholarly journals The minimum and the primitive representation of positive definite quadratic forms

1994 ◽  
Vol 133 ◽  
pp. 127-153 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Primitive Representation ◽  
Quadratic Lattices

Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. When there is an isometry from M to N, we say that M is represented by N (even in the local cases). In the following, we assume that the localization Mp is represented by Np for every prime p. Let us consider the following assertion Am,n(N):Am,n(N): There exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

scholarly journals The minimum and the primitive representation of positive definite quadratic forms II

1996 ◽  
Vol 141 ◽  
pp. 1-27 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Primitive Representation ◽  
Quadratic Lattices

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion Am, n : Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. We assume that the localization Mp is represented by Np for every prime p, that is there is an isometry from Mp to Np. Then there exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

scholarly journals Tensor products of positive definite quadratic forms IV

1979 ◽  
Vol 73 ◽  
pp. 149-156 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Orthogonal Groups ◽  
Quadratic Lattices

Let L, M, N be positive definite quadratic lattices over Z. We treated the following problem in [5], [6]:If L ⊗ M is isometric to L ⊗ N, then is M isometric to N?We gave a condition (**) in [6] such that the answer is affirmative for an indecomposable lattice L satisfying (**), and we gave some examples. In this paper we introduce a certain apparently weaker condition (A) than the condition (**), and we show that the condition (A) implies the condition (**) and more on integral orthogonal groups than a result in [6].

scholarly journals Tensor Products of Positive definite Quadratic forms III

1978 ◽  
Vol 70 ◽  
pp. 173-181 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
List Type ◽  
Quadratic Lattices

In the previous papers [2], [3] we treated the following two questions. Let L,M,N be positive definite quadratic lattices over Z: (i) If L, M are indecomposable, then is L⊗M indecomposable?(ii) Does L ⊗ M ⋍ L ⊗ N imply M ⋍ N?

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

Local deformations of positive-definite quadratic forms

2012 ◽  
Vol 64 (7) ◽  
pp. 1019-1035
Author(s):  
V. M. Bondarenko ◽  
V. V. Bondarenko ◽  
Yu. N. Pereguda
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Local Deformations

scholarly journals Gaps between values of positive definite quadratic forms

1972 ◽  
Vol 22 (1) ◽  
pp. 87-105 ◽  
Author(s):  
H. Davenport ◽  
D. Lewis
Keyword(s):  
Quadratic Forms ◽  
Positive Definite
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