共查询到20条相似文献,搜索用时 15 毫秒
1.
Zhi-Hong Sun 《Journal of Number Theory》2007,124(1):62-104
Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u2−dv2 and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. 相似文献
2.
Zhi-Hong Sun 《Journal of Number Theory》2009,129(3):499-550
Let be a prime and a,b∈Z with a2+b2≠p. Suppose p=x2+(a2+b2)y2 for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U0=0, U1=1 and Un+1=bUn+k2Un−1(n?1). We also pose many conjectures concerning , or . 相似文献
3.
Zhi-Hong Sun 《Journal of Number Theory》2008,128(5):1295-1335
Let be a prime. Let a,b∈Z with p?a(a2+b2). In the paper we mainly determine by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with AC−B2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1(n?1). 相似文献
4.
5.
T. M. K. Davison 《Aequationes Mathematicae》1987,34(1):78-81
Ohne Zusammenfassung
Dedicated to Professor Otto Haupt with best wishes on his 100th birthday 相似文献
6.
Mei-Chu Chang 《Journal of Number Theory》2009,129(9):2064-2071
We establish character sum bounds of the form
7.
Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ2Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ2S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V3 into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ3T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation
F(t)+t3G(1/t)=0, 相似文献
8.
Steve Wright 《Journal of Number Theory》2007,123(1):120-132
If S is a nonempty, finite subset of the positive integers, we address the question of when the elements of S consist of various mixtures of quadratic residues and nonresidues for infinitely many primes. We are concerned in particular with the problem of characterizing those subsets of integers that consist entirely of either (1) quadratic residues or (2) quadratic nonresidues for such a set of primes. We solve problem (1) and we show that problem (2) is equivalent to a purely combinatorial problem concerning families of subsets of a finite set. For sets S of (essentially) small cardinality, we solve problem (2). Related results and some associated enumerative combinatorics are also discussed. 相似文献
9.
Ay?e Alaca 《Journal of Number Theory》2011,131(11):2192-2218
Explicit formulae are determined for the number of representations of a positive integer by the quadratic forms ax2+by2+cz2+dt2 with a,b,c,d∈{1,4,9,36}, gcd(a,b,c,d)=1 and a?b?c?d. 相似文献
10.
Svetozar Kurepa 《Aequationes Mathematicae》1987,34(2-3):125-138
The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday. 相似文献
11.
Given an arbitrary n, we consider anisotropic quadratic forms of dimension n over all fields of characteristic ≠ 2 and prove that the height of an n-dimensional excellent form (depending on n only) is the (precise) lower bound of the heights of all forms of dimension n.
The second and third authors were partially supported by the European Community’s Human Potential Programme under contract
HPRN-CT-2002-00287, KTAGS. The James D.Wolfensohn Fund and The Ellentuck Fund support is acknowledged by the second author.
Received: 9 December 2005 相似文献
12.
Martin ech Dominik Lachman Josef Svoboda Magdalna Tinkov Kristýna Zemkov 《Mathematische Nachrichten》2019,292(3):540-555
The aim of this article is to study (additively) indecomposable algebraic integers of biquadratic number fields K and universal totally positive quadratic forms with coefficients in . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field K. Furthermore, estimates are proven which enable algorithmization of the method of escalation over K. These are used to prove, over two particular biquadratic number fields and , a lower bound on the number of variables of a universal quadratic forms. 相似文献
13.
D. W. Lewis 《Linear and Multilinear Algebra》1997,42(4):349-363
A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms. 相似文献
14.
Oleg T. Izhboldin 《manuscripta mathematica》2000,102(1):41-52
Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X
φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X
φ and X
ψ coincide but . For a pair of anisotropic (2
n
-1)-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X.
Received: 27 September 1999 / Revised version: 27 December 1999 相似文献
15.
A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004 相似文献
16.
A.S. Sivatski 《Journal of Pure and Applied Algebra》2018,222(3):560-567
Let F be a field of characteristic distinct from 2, a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, , their matrices. We say that the pair is a k-pair if there exist such that all the entries of the upper-left corner of the matrices and are in F. We give certain criteria to determine whether a given pair is a k-pair. We consider the transfer determined by the -linear map with , , and prove that if , then is a -pair. If, additionally, the form does not have a totally isotropic subspace of dimension over , we show that is a -pair. In particular, if the form is anisotropic, and , then is a k-pair. 相似文献
17.
Zhi-Wei Sun 《Journal of Number Theory》2011,131(12):2387-2397
The nth Delannoy number and the nth Schröder number given by
18.
Stanislav Popovych 《Linear algebra and its applications》2010,433(1):164-171
Let
19.
Venkata Balaji Thiruvalloor Eesanaipaadi 《Journal of Pure and Applied Algebra》2007,208(1):237-259
We study degenerations of rank 3 quadratic forms and of rank 4 Azumaya algebras, and extend what is known for good forms and Azumaya algebras. By considering line-bundle-valued forms, we extend the theorem of Max-Albert Knus that the Witt-invariant—the even Clifford algebra of a form—suffices for classification. An algebra Zariski-locally the even Clifford algebra of a ternary form is so globally up to twisting by square roots of line bundles. The general, usual and special orthogonal groups of a form are determined in terms of automorphism groups of its Witt-invariant. Martin Kneser’s characteristic-free notion of semiregular form is used. 相似文献
20.
We prove Witt’s cancelation and extension theorems for Galois Ring valued quadratic forms. The proof is based on the properties of the invariant I, previously defined by the authors, that classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic forms. Our results extend the Witt’s theorem for mod four valued quadratic forms. On the other hand, the known relation between the invariant I and the Arf invariant of an ordinary quadratic form (if the associated nonsingular bilinear form is alternating) is extended to the nonalternating case by explaining the invariant I in terms of Clifford algebras. 相似文献