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1.
As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms. 相似文献
2.
A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger. 相似文献
3.
S. A. Gritsenko 《Mathematical Notes》2007,81(1-2):172-182
In this paper, we solve Goldbach’s ternary problem involving primes expressible by given primitive positive definite binary quadratic forms whose discriminants coincide with the discriminants of imaginary quadratic fields in which quadratic forms split into linear multipliers. 相似文献
4.
The Ramanujan Journal - 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... 相似文献
5.
A. V. Khorosheva 《Journal of Mathematical Sciences》2003,118(1):4933-4942
We study branching of representations of a locally p-one-dimensional form by a genus of positive definite integral quadratic forms. We give a complete list of minimal representations by a genus for forms of square level. GaussMinkowski formulas are obtained for heights of representations over the ring of integers. As an application, we obtain formulas for heights of primitive representations by genera for specific forms constructed by the method of orthogonal complement. Bibliography: 6 titles. 相似文献
6.
Byeong-Kweon Oh 《Mathematische Zeitschrift》2003,244(2):399-413
In this article, we prove that there are only finitely many positive definite integral quadratic forms of rank n+3(n≥2) that represent all positive definite integral quadratic forms of rank n but finitely many exceptions. Furthermore we determine all diagonal quadratic forms having such property and its exceptions
remaining four as candidates.
Received: 29 November 2000 ; in final form: 8 August 2002 /
Published online: 1 April 2003
Mathematics Subject Classification (2000): 11E12, 11E20. 相似文献
7.
On the Construction of Nondecomposable Definite Hermitian Forms 总被引:1,自引:0,他引:1
51.Introducti0nLetF=Q(In)beanimaginaryquadraticfieldandD.theringofintegersofF.wheremisasquare-freepositiveinteger.ForapositivedefiniteD.-latticeLthesetSesq>.(L)ofpositivesemidefinitesesquilinearformsoverD.,0requivalently,thesetofpositivesemidefiniteHermitianmatrices0verD.isasemigroupunderadd1tlon.AformoverD.whichcanbewrittenasthesumoftwon0n-zer0elementsinSesq>,(L)iscalledadditivelydecomposable,otherwlseitiscallednondecomp0sab1e.IftheD.-lattice(L,9)istheorthogonalsumoftwonon-zerosublatti… 相似文献
8.
Young Min Lee 《The Ramanujan Journal》2008,16(1):97-104
In this paper, we find all quaternary universal positive definite integral quadratic forms over
and prove an analogue of Conway and Schneeberger’s 15-Theorem.
This work was partially supported by KRF(2003-070-C00001).
This work was supported by the Brain Korea 21 project in 2004. 相似文献
9.
The Ramanujan Journal - In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for... 相似文献
10.
Charles J Parry 《Journal of Number Theory》1973,5(4):266-270
In 1882 Weber showed that any primitive binary quadratic form with integral coefficients represents infinitely many primes in any arithmetic progression consistent with the generic characters of the form. In this paper it is shown that for any two primitive integral binary quadratic forms with unequal but fundamental discriminants, there is an infinite set of prime numbers p in any arithmetic progression consistent with the generic characters of the forms such that both forms represent p. 相似文献
11.
Zhu Fuzu 《数学年刊B辑(英文版)》1994,15(3):349-360
CONSTRUCTIONOFINDECOMPOSABLEDEFINITEHERMITIANFORMS¥ZHUFUZU(DepartmelltofMathematics,EastChinaNormalUniversitytShanghai200062,... 相似文献
12.
Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms
Byeong Moon Kim 《manuscripta mathematica》1999,99(2):181-184
In this paper, we will prove if D is large enough, there are no positive integral diagonal septanary universal quadratic forms over .
Received: 13 November 1997 / Revised version: 17 November 1998 相似文献
13.
Byeong-Kweon Oh 《Inventiones Mathematicae》2007,170(2):421-453
A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases. 相似文献
14.
The Ramanujan Journal - Let m, n be positive integers with $$m\le n$$ . Let $$\kappa (m,n)$$ be the largest integer k such that for any (positive definite and integral) quadratic forms... 相似文献
15.
Wenguang ZHAI 《Frontiers of Mathematics in China》2022,17(6):1089
Let l > 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in l variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u3 = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work. 相似文献
16.
Jesse I. Deutsch 《Archiv der Mathematik》2008,91(1):44-48
In a paper of Kim, Chan, and Rhagavan, the universal ternary classical quadratic forms over quadratic fields of positive discriminant
were discovered. Here a proof of the universality of some of these quadratic forms is given using a technique of Liouville.
Another quadratic form over the field of discriminant 8 is shown universal by a different elementary approach.
Received: 30 October 2007 相似文献
17.
S. V. Fedorova 《Journal of Mathematical Sciences》2003,118(1):4895-4903
We study the branching of representations of a p-elementary quadratic form by a genus of positive definite locally p-two-dimensional forms. A primitive representation of a p-elementary form is decomposed into a direct sum of minimal indecomposable representations; the latter representations are found in an explicit form. For the case of branching, we find local multipliers of the weight of representations of a form by a genus. As an application, we calculate the number of embeddings into the classical root lattices. The method of orthogonal complement is applied in constructing new genera of quadratic forms. Bibliography: 9 titles. 相似文献
18.
V. L. Kulik 《Ukrainian Mathematical Journal》1990,42(12):1548-1552
The problem of the existence of quadratic forms that have a positive definite derivative along the solutions of linear extensions of dynamical systems on a torus is considered. Assuming the existence of quadratic forms whose derivative is positive definite only with respect to part of the variables, conditions ensuring the existence of a quadratic form whose derivative is already positive definite with respect to all variables are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1713–1717, December, 1990. 相似文献
19.
For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S
a
={an
2∣n≥2} or S=S
a,b
={an
2+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S
a
-universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S
1-universal quadratic forms not representing 1. 相似文献
20.
F. A. Andrianov 《Journal of Mathematical Sciences》1997,83(6):694-719
Multiplicative dependence of integral representations of integers by positive definite quadratic forms in an odd number of
variables on square factors of the represented numbers is studied.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 15–55.
Translated by A. S. Goloubeva. 相似文献