Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula

1996 ◽  
pp. 79-104 ◽  
Author(s):  
E. Becker ◽  
J. P. Cardinal ◽  
M.-F. Roy ◽  
Z. Szafraniec
Keyword(s):  
Kronecker Symbol

scholarly journals The fourth dimension subgroups and polynomial maps, II

1978 ◽  
Vol 69 ◽  
pp. 1-7 ◽  
Author(s):  
Ken-Ichi Tahara
Keyword(s):  
Linear Equations ◽  
Integral Solution ◽  
Fourth Dimension ◽  
Direct Products ◽  
Kronecker Symbol ◽  
Cyclic Groups ◽  
Polynomial Maps ◽  
Polynomial Map

In our previous paper [3] we proved the following ([3, Theorem 16]) :THEOREM A. Let G be a 2-group of class 3. Let G2 and G/G2 be direct products of cyclic groups 〈yq〉 of order αq (1 ≦ q ≦ m), and of cyclic groups 〈hi〉 of order βi (1 ≦ i ≦ n) with β1 ≧ β2 ≧ · · · βn, respectively. Let xi be representatives of hi (1 ≦ i ≦ n), and put Then a homomorphism ψ:G3→T can be extended to a polynomial map from G to T of degree ≦ 4 if and only if there exists an integral solution in the following linear equations of Xiq (1 ≦ i ≦ n, 1 ≦ q ≦ m) with coefficients in T: (I)where δij is the Kronecker symbol for βi: i.e. δij = 1 or 0 according to βi = βj or βi > βj, respectively.

scholarly journals Class Numbers and Biquadratic Reciprocity

1982 ◽  
Vol 34 (4) ◽  
pp. 969-988 ◽  
Author(s):  
Kenneth S. Williams ◽  
James D. Currie
Keyword(s):  
Real Number ◽  
Class Number ◽  
Quadratic Field ◽  
Class Numbers ◽  
Kronecker Symbol ◽  
The Real ◽  
Unique Manner ◽  
Image Position

0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,(0.1)with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by(0.2)Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by(0.3)These choices are assumed throughout.The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].

An Upper Bound on the Least Inert Prime in a Real Quadratic Field

2000 ◽  
Vol 52 (2) ◽  
pp. 369-380 ◽  
Author(s):  
Andrew Granville ◽  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Upper Bound ◽  
Quadratic Field ◽  
Computational Techniques ◽  
Real Quadratic Field ◽  
Kronecker Symbol ◽  
Fundamental Discriminant ◽  
Real Quadratic

AbstractIt is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D > 3705, there is always at least one prime p < √D/2 such that the Kronecker symbol (D/p) = −1.

scholarly journals Small zeros of quadratic L-functions

1993 ◽  
Vol 47 (2) ◽  
pp. 307-319 ◽  
Author(s):  
Ali E. Özlük ◽  
C. Snyder
Keyword(s):  
Asymptotic Behaviour ◽  
Real Axis ◽  
Riemann Hypothesis ◽  
Kronecker Symbol ◽  
Absolute Value ◽  
The Real ◽  
Generalized Riemann Hypothesis ◽  
Image Position

We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that |K(1/2 ± it)| is rapidly decreasing as t increases. We investigate the asymptotic behaviour ofas D → ∞. Here denotes the sum over the non-trivial zeros p = 1/2 + iγ of the Dirichlet L-function L(s, χd), and χd = () is the Kronecker symbol. The outer sum is over all fundamental discriminants d that are in absolute value ≤ D. Assuming the Generalized Riemann Hypothesis, we show that for

scholarly journals Inequalities for inert primes and their applications

2020 ◽  
Vol 16 (08) ◽  
pp. 1819-1832
Author(s):  
Zilong He
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Kronecker Symbol ◽  
Ternary Quadratic Forms ◽  
New Criterion

For any given non-square integer [Formula: see text], we prove Euclid’s type inequalities for the sequence [Formula: see text] of all primes satisfying the Kronecker symbol [Formula: see text], [Formula: see text] and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones’s argument in the classification of regular ternary quadratic forms to some extent.

scholarly journals A Unit Norm Conjecture for Some Real Quadratic Number Fields: A Preliminary Heuristic Investigation

2020 ◽  
pp. 46-50
Author(s):  
Elliot Benjamin
Keyword(s):  
Probability Density ◽  
Number Field ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Kronecker Symbol ◽  
Theoretical Assumptions ◽  
Quadratic Number Fields ◽  
Real Quadratic ◽  
Unit Norm

In this paper we make a conjecture about the norm of the fundamental unit, N(e), of some real quadratic number fields that have the form k = Q(√(p1.p2) where p1 and p2 are distinct primes such that pi = 2 or  pi ≡ 1 mod 4, i = 1, 2. Our conjecture involves the case where the Kronecker symbol (p1/p2) = 1 and the biquadratic residue symbols (p1/p2)4 = (p2/p1)4 = 1, and is based upon Stevenhagen’s conjecture that if k = Q(√(p1.p2) is any real quadratic number field as above, then P(N(e) = -1)) = 2/3, i.e., the probability density that N(e) = -1 is 2/3. Given Stevenhagen’s conjecture and some theoretical assumptions about the probability density of the Kronecker symbols and biquadratic residue symbols, we establish that if k is as above with (p1/p2) = (p1/p2)4 = (p2/p1)4 = 1, then P(N(e) = -1)) = 1/3, and we support our conjecture with some preliminary heuristic data.

scholarly journals A NOTE ON SPECIAL VALUES OF CERTAIN DIRICHLET L-FUNCTIONS

2010 ◽  
Vol 83 (3) ◽  
pp. 435-438
Author(s):  
B. RAMAKRISHNAN
Keyword(s):  
Modular Forms ◽  
Direct Consequence ◽  
Kronecker Symbol ◽  
Special Values ◽  
Half Integral Weight ◽  
Divisor Functions ◽  
Integral Weight ◽  
Finite Sums

AbstractIn Gun and Ramakrishnan [‘On special values of certain Dirichlet L-functions’, Ramanujan J.15 (2008), 275–280], we gave expressions for the special values of certain Dirichlet L-function in terms of finite sums involving Jacobi symbols. In this note we extend our earlier results by giving similar expressions for two more special values of Dirichlet L-functions, namely L(−1,χm) and L(−2,χ−m′), where m,m′ are square-free integers with m≡1 mod 8 and m′≡3 mod 8 and χD is the Kronecker symbol $(\frac {D}{\cdot })$. As a consequence, using the identities of Cohen [‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann.217 (1975), 271–285], we also express the finite sums with Jacobi symbols in terms of sums involving divisor functions. Finally, we observe that the proof of Theorem 1.2 in Gun and Ramakrishnan (as above) is a direct consequence of Equation (24) in Gun, Manickam and Ramakrishnan [‘A canonical subspace of modular forms of half-integral weight’, Math. Ann.347 (2010), 899–916].

scholarly journals A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Garcia Villeda
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Dirichlet Character ◽  
Imaginary Quadratic Field ◽  
Kronecker Symbol ◽  
Quadratic Residues ◽  
Elementary Methods ◽  
Computable Formula

<p style='text-indent:20px;'>Using elementary methods, we count the quadratic residues of a prime number of the form <inline-formula><tex-math id="M2">\begin{document}$ p = 4n-1 $\end{document}</tex-math></inline-formula> in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula> of the imaginary quadratic field <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Q(\sqrt{-p}). $\end{document}</tex-math></inline-formula> Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.</p>

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