On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.

Download Full-text

On the realizable classes of the square root of the inverse different in the unitary class group

International Journal of Number Theory ◽

10.1142/s1793042117500476 ◽

2017 ◽

Vol 13 (04) ◽

pp. 913-932 ◽

Cited By ~ 1

Author(s):

Sin Yi Cindy Tsang

Keyword(s):

Abelian Group ◽

Number Field ◽

Bilinear Form ◽

Class Group ◽

Finite Abelian Group ◽

Symmetric Bilinear Form ◽

Square Root ◽

Ring Of Integers ◽

Trace Map ◽

Odd Order

Let [Formula: see text] be a number field with ring of integers [Formula: see text] and let [Formula: see text] be a finite abelian group of odd order. Given a [Formula: see text]-Galois [Formula: see text]-algebra [Formula: see text], write [Formula: see text] for its trace map and [Formula: see text] for its square root of the inverse different, where [Formula: see text] exists by Hilbert’s formula since [Formula: see text] has odd order. The pair [Formula: see text] is locally [Formula: see text]-isometric to [Formula: see text] whenever [Formula: see text] is weakly ramified, in which case it defines a class in the unitary class group [Formula: see text] of [Formula: see text]. Here [Formula: see text] denotes the canonical symmetric bilinear form on [Formula: see text] defined by [Formula: see text] for all [Formula: see text]. We will study the set of all such classes and show that a subset of them forms a subgroup of [Formula: see text].

Download Full-text

SMALL ZEROS OF QUADRATIC FORMS OVER $\overline{\mathbb{Q}}$

International Journal of Number Theory ◽

10.1142/s1793042108001481 ◽

2008 ◽

Vol 04 (03) ◽

pp. 503-523 ◽

Cited By ~ 2

Author(s):

LENNY FUKSHANSKY

Keyword(s):

Quadratic Form ◽

Quadratic Forms ◽

Number Fields ◽

Dimensional Subspace ◽

Orthogonal Decomposition ◽

Isotropic Subspace ◽

Effective Version ◽

Small Height ◽

Totally Isotropic Subspace

Let N ≥ 2 be an integer, F a quadratic form in N variables over [Formula: see text], and [Formula: see text] an L-dimensional subspace, 1 ≤ L ≤ N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over [Formula: see text] of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over [Formula: see text]. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over [Formula: see text]. This extends previous results of the author over number fields. All bounds on height are explicit.

Download Full-text

Matroid connectivity and singularities of configuration hypersurfaces

Letters in Mathematical Physics ◽

10.1007/s11005-020-01352-3 ◽

2021 ◽

Vol 111 (1) ◽

Author(s):

Graham Denham ◽

Mathias Schulze ◽

Uli Walther

Keyword(s):

Bilinear Form ◽

Symmetric Bilinear Form ◽

Smooth Locus

AbstractConsider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

Download Full-text

Supervised learning using a symmetric bilinear form for record linkage

Information Fusion ◽

10.1016/j.inffus.2014.11.004 ◽

2015 ◽

Vol 26 ◽

pp. 144-153 ◽

Cited By ~ 7

Author(s):

Daniel Abril ◽

Vicenç Torra ◽

Guillermo Navarro-Arribas

Keyword(s):

Supervised Learning ◽

Bilinear Form ◽

Record Linkage ◽

Symmetric Bilinear Form

Download Full-text

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-160-x ◽

2012 ◽

Vol 55 (2) ◽

pp. 418-423 ◽

Cited By ~ 3

Author(s):

Le Anh Vinh

Keyword(s):

Positive Integer ◽

Finite Field ◽

Finite Fields ◽

Bilinear Form ◽

Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

Download Full-text

Trace Forms on Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-046-5 ◽

1962 ◽

Vol 14 ◽

pp. 553-564 ◽

Cited By ~ 14

Author(s):

Richard Block

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Bilinear Form ◽

Symmetric Bilinear Form ◽

Trace Form ◽

Finite Degree ◽

Trace Forms ◽

Matrix Products ◽

The Matrix ◽

Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Download Full-text

Теплопроводность цепочки ротаторов с двухбарьерным потенциалом взаимодействия

Физика твердого тела ◽

10.21883/ftt.2021.07.51051.016 ◽

2021 ◽

Vol 63 (7) ◽

pp. 975

Author(s):

А.П. Клинов ◽

М.А. Мазо ◽

В.В. Смирнов

Keyword(s):

Thermal Conductivity ◽

Heat Conductivity ◽

Normal Modes ◽

Double Barrier ◽

One Dimensional ◽

Internal Barrier ◽

Local Fluctuations ◽

Small Height ◽

Coefficient Of Thermal Conductivity ◽

Height Dependence

The thermal conductivity of a one-dimensional chain of rotators with a double-barrier interaction potential of nearest neighbors has been studied numerically. We show that the height of the "internal" barrier, which separates topologically nonequivalent degenerate states, significantly affects the temperature dependence of the heat conductivity of the system. The small height of this barrier leads to the dominant contribution of the non-linear normal modes at low temperatures. In such a case the coefficient of thermal conductivity turns out to be the risen function of the temperature. The growth of the coefficient is limited by local fluctuations corresponding to jumps over the barriers. At higher values of the internal barrier height, dependence of the heat conductivity on temperature is similar to that of classical rotators.

Download Full-text