A Nontrivial Zero.: 11020

2004 ◽  
Vol 111 (4) ◽  
pp. 366
Author(s):  
M. L. Glosser ◽  
J. A. Grzesik
Keyword(s):  
Nontrivial Zero

scholarly journals On systems of diagonal forms

2007 ◽  
Vol 82 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Michael P. Knapp
Keyword(s):  
Integer Coefficients ◽  
Diagonal Forms ◽  
Nontrivial Zero

AbstractIn this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field Qp provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in Qp. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.

NORMAL PAIRS WITH ZERO-DIVISORS

2011 ◽  
Vol 10 (02) ◽  
pp. 335-356 ◽  
Author(s):  
DAVID E. DOBBS ◽  
JAY SHAPIRO
Keyword(s):  
Prime Ideal ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Valuation Domain ◽  
Ring Extension ◽  
Infinite Products ◽  
Normal Pair ◽  
Zero Divisors ◽  
Pair Property ◽  
Nontrivial Zero

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

Exact values of Γ∗(10,p)

2019 ◽  
Vol 16 (03) ◽  
pp. 639-649
Author(s):  
Daiane S. Veras ◽  
Paulo H. A. Rodrigues
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Diagonal Form ◽  
Integer Coefficients ◽  
Nontrivial Zero ◽  
Special Case

For [Formula: see text] and [Formula: see text] a prime number, define [Formula: see text] to be the smallest positive integer [Formula: see text] such that any diagonal form [Formula: see text], with integer coefficients, has nontrivial zero over [Formula: see text] whenever [Formula: see text]. A special case of a conjecture attributed to Artin states that [Formula: see text]. It is well known that the equality occurs when [Formula: see text]. In this paper, we obtain the exact values of [Formula: see text] for all primes [Formula: see text] and, except for [Formula: see text], these values are much lower than those established in the conjecture, as might be expected.

scholarly journals On systems of diagonal forms II

2009 ◽  
Vol 147 (1) ◽  
pp. 31-45 ◽  
Author(s):  
MICHAEL P. KNAPP
Keyword(s):  
Diagonal Forms ◽  
Nontrivial Zero

AbstractGiven a system of diagonal forms over ℚp, we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies p > (largest degree) − (smallest degree) + 1, then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms.

The u-invariant of p-adic function fields

2013 ◽  
Vol 2013 (679) ◽  
pp. 65-73 ◽  
Author(s):  
David B. Leep
Keyword(s):  
Quadratic Form ◽  
Recent Result ◽  
Quadratic Forms ◽  
Function Fields ◽  
Field Extension ◽  
Finitely Generated ◽  
Nontrivial Zero

Abstract Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.

scholarly journals On continuity of derivations and epimorphisms on some vector-valued group algebras

1997 ◽  
Vol 56 (2) ◽  
pp. 209-215
Author(s):  
Ramesh V. Garimella
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Group Algebras ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Zero Divisors ◽  
Integrable Functions ◽  
Vector Valued ◽  
Nontrivial Zero

For a locally compact Abelian group G and a commutative Banach algebra B, let L1(G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is compact and B is a nonunital Banach algebra without nontrivial zero divisors, then (i) all derivations on L1(G, B) are continuous if and only if all derivations on B are continuous, and (ii) each epimorphism from a Banach algebra X onto L1(G, B) is continuous provided every epimorphism from X onto B is continuous. If G is noncompact then every derivation on L1(G, B) and every epimorphism from a commutative Banach algebra onto L1(G, B) are continuous. Our results extend the results of Neumann and Velasco for nonunital Banach algebras.

scholarly journals A zero-dimensional topologically nontrivial state in a superconducting quantum dot

2018 ◽  
Vol 9 ◽  
pp. 1705-1714 ◽  
Author(s):  
Pasquale Marra ◽  
Alessandro Braggio ◽  
Roberta Citro
Keyword(s):  
Quantum Dot ◽  
Topological States Of Matter ◽  
Finite Size ◽  
Current Phase ◽  
Discrete Energy ◽  
Topological Superconductor ◽  
Topological States ◽  
The One ◽  
Nontrivial Zero

The classification of topological states of matter in terms of unitary symmetries and dimensionality predicts the existence of nontrivial topological states even in zero-dimensional systems, i.e., systems with a discrete energy spectrum. Here, we show that a quantum dot coupled with two superconducting leads can realize a nontrivial zero-dimensional topological superconductor with broken time-reversal symmetry, which corresponds to the finite size limit of the one-dimensional topological superconductor. Topological phase transitions corresponds to a change of the fermion parity, and to the presence of zero-energy modes and discontinuities in the current–phase relation at zero temperature. These fermion parity transitions therefore can be revealed by the current discontinuities or by a measure of the critical current at low temperatures.

A CHARACTERIZATION OF NONSINGULAR PAIRS OF QUADRATIC FORMS

2002 ◽  
Vol 01 (04) ◽  
pp. 391-412 ◽  
Author(s):  
DAVID B. LEEP ◽  
LAURA MANN SCHUELLER
Keyword(s):  
Quadratic Forms ◽  
Algebraic Closure ◽  
Arbitrary Field ◽  
Arf Invariant ◽  
Nontrivial Zero

Let F, G be a pair of quadratic forms defined over an arbitrary field k. We give a characterization for when every nontrivial zero of F = G = 0 defined over the algebraic closure of k is nonsingular. When chark ≠ 2, this result is well known. When chark = 2, the problem divides into two cases. If n is odd, we use the half-determinant, and if n is even, we use the Arf invariant for this characterization. The characterization depends only on the coefficients of the quadratic forms and operations taking place in the field k.

A Note on Artin's Diophantine Conjecture

1970 ◽  
Vol 13 (1) ◽  
pp. 119-120
Author(s):  
George Maxwell
Keyword(s):  
Quadratic Form ◽  
Nontrivial Zero

A well known theorem of Hasse [1] says that every quadratic form in at least 5 variables over the field Qp of p-adic numbers has a nontrivial zero. This fact has led Artin to make the conjecture(C): "Every form over Qp of degree d in n > d2 variables has a non-trivial zero." However, a counterexample has been provided by Terjanian [2] in the case d=4.The case d=2 is distinguished by the fact that every quadratic form may be "diagonalized", i.e., assumed to be of the type Σ aiX2i. One is therefore led to the weaker conjecture(C): "Every form f= Σ aiXdi over Qp in n > d2 variables has a nontrivial zero in Qp,"which still generalizes Hasse's theorem.

scholarly journals Normal characterizations of lattices

2001 ◽  
Vol 28 (10) ◽  
pp. 561-570
Author(s):  
Carmen D. Vlad
Keyword(s):  
Topological Properties ◽  
Finitely Additive Measures ◽  
Additive Measures ◽  
Nontrivial Zero

LetXbe an arbitrary nonempty set andℒa lattice of subsets ofXsuch that∅,X∈ℒ. Let𝒜(ℒ)denote the algebra generated byℒandI(ℒ)denote those nontrivial, zero-one valued, finitely additive measures on𝒜(ℒ). In this paper, we discuss some of the normal characterizations of lattices in terms of the associated lattice regular measures, filters and outer measures. We consider the interplay between normal lattices, regularity orσ-smoothness properties of measures, lattice topological properties and filter correspondence. Finally, we start a study of slightly, mildly and strongly normal lattices and express then some of these results in terms of the generalized Wallman spaces.

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