We investigate the problem of r almost-primes represented by sets of quadratic forms and give upper bounds for r. Our results extend work of Diamond and Halberstam in which they ā¦
We investigate the problem of r almost-primes represented by sets of quadratic forms and give upper bounds for r. Our results extend work of Diamond and Halberstam in which they investigated the corresponding problem for polynomials.
Let k be a field of characteristic distinct from 2, V a finite dimensional vector space over k. We call two pairs of quadratic k-forms (f1,g1), (f2,g2) on V isomorphic ā¦
Let k be a field of characteristic distinct from 2, V a finite dimensional vector space over k. We call two pairs of quadratic k-forms (f1,g1), (f2,g2) on V isomorphic if there exists an isomorphism s:VāV such that f2=f1ās, g2=g1ās. We prove that if f1+tg1āf2+tg2 over k(t) and either the form f1+tg1 is anisotropic, or det(f1+tg1) is a squarefree polynomial of degree at least dimVā1, then the pairs (f1,g1) and (f2,g2) are isomorphic.
Let s be a natural number, s ā„ 2. We seek a positive number Ī»( s ) with the following property: Let Īµ > 0. Let Q 1 (x 1 ā¦
Let s be a natural number, s ā„ 2. We seek a positive number Ī»( s ) with the following property: Let Īµ > 0. Let Q 1 (x 1 , ā¦, x s ), Q 2 (x 1 , ā¦, x s ) be real quadratic forms, then for N > C 1 ( s , Īµ) we have for some integers n 1 , ā¦, n s ,
The linkage between abuse to artisanal cobalt minersāincluding childrenāin the Democratic Republic of the Congo (DRC) and use of cobalt in advanced batteries has prompted global supply chain reviews, responsible ā¦
The linkage between abuse to artisanal cobalt minersāincluding childrenāin the Democratic Republic of the Congo (DRC) and use of cobalt in advanced batteries has prompted global supply chain reviews, responsible sourcing initiatives, and ...From 2000 through 2020, demand for cobalt to manufacture batteries grew 26-fold. Eighty-two percent of this growth occurred in China and China's cobalt refinery production increased 78-fold. Diminished industrial cobalt mine production in the early-to-mid ...
Quadratic forms of height two and leading form defined over the base field are determined over several fields.Also forms of height and degree two over an arbitrary field are classified.Knebusch, ā¦
Quadratic forms of height two and leading form defined over the base field are determined over several fields.Also forms of height and degree two over an arbitrary field are classified.Knebusch, in his theory of generic splitting fields, defines the height of a quadratic form q over a field 7 of characteristic not 2. A form q has height 1 if a splits over F(q), where F(q) denotes the function field of the projective variety q = 0, and a form q has height n > 1 if, over F(q),q has height n -1.The (even-dimensional) forms of height 1 are Pfister forms and their scalar multiples.This is a restatement of the well-known fact that Pfister forms are the only (anisotropic) strongly multiplicative forms.We consider here forms of height 2 and obtain a classification in several cases.Forms of height 2 are of interest because they are the simplest possible forms aside from Pfister forms.Nonetheless, such forms remain somewhat mysterious, and a complete classification appears to be extremely difficult.This paper extends Knebusch's work, particularly the last two sections of [13].From the remarks above, we have that (even-dimensional) q has height 2 if and only if ker(a Ā® 7(a)) is a scalar times some Pfister form p. The degree of q is n if p is an n-fold Pfister form (or 0 if q is odd dimensional).Particular attention is given to the case where p is defined over 7-a is then called a good form.Knebusch has shown any form of degree 1 and height 2 is good and has classified these.However, height 2 forms need not in general be good.Ā§1 studies good forms of height 2. Such forms of degrees 0 and 2 are classified.By placing relatively strong restrictions on F, all good forms of height 2 may be determined.In particular, there is a classification for fields of transcendence degree ^ 4 over C, transcendence degree < 2 over R, global fields and Q((t)).Ā§2 considers forms of height and degree 2. Although we do not obtain a complete description (except in a few cases), such forms are shown to be one of five types.This is sufficient for some apphcations.For example, if q has height and degree 2 then the Witt Kernel W(F(q)/F) is a Pfister ideal.Notation and terminology will be taken from [15], except we will writeker(a) for the anisotropic part of a form a. The discriminant of a will be denoted by d(q); thus d(q) = (-l)"("-1)/2det(a), where n = dim a.
Quadratic forms of height two and leading form defined over the base field are determined over several fields. Also forms of height and degree two over an arbitrary field are ā¦
Quadratic forms of height two and leading form defined over the base field are determined over several fields. Also forms of height and degree two over an arbitrary field are classified.
Let [Formula: see text] be a field of characteristic 2. In this paper, we provide an interesting application of quadratic forms over [Formula: see text] in determination of the Wedderburn ā¦
Let [Formula: see text] be a field of characteristic 2. In this paper, we provide an interesting application of quadratic forms over [Formula: see text] in determination of the Wedderburn decomposition of the rational group algebra [Formula: see text], where [Formula: see text] is a real special [Formula: see text]-group. We further apply these computations to exhibit two non-isomorphic real special [Formula: see text]-groups with isomorphic rational group algebra.
The Grothendieck group of finite-length inner product modules over a PID is here shown to be a sum of countably many copies of the corresponding groups for the residue fields.It ā¦
The Grothendieck group of finite-length inner product modules over a PID is here shown to be a sum of countably many copies of the corresponding groups for the residue fields.It follows that nonsingular pairs of symmetric bilinear forms in characteristic 2 owe their extra complexity only to lack of a cancellation theorem: The invariants for isometry in other characteristics continue to determine classes in the Grothendieck group.This is also true for singular pairs. 1* Inner products on finite-length modules*Let R be a
In this article, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an ā¦
In this article, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given by Henryk Iwaniec and Ritabrata Munshi in \cite{H-R}.