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quadratic form

Contents

Definition

For VV a vector space or more generally a kk-module, then a quadratic form on VV is a function

q:Vk q\colon V \to k

such that for all vVv \in V, tkt \in k

q(tv)=t 2q(v) q(t v) = t^2 q(v)

and the polarization of qq

(v,w)q(v+w)q(v)q(w) (v,w) \mapsto q(v+w) - q(v) - q(w)

is a bilinear form.

Quadratic refinement

Let

,:VVk \langle -,-\rangle \colon V \otimes V \to k

be a bilinear form. A function

q:Vk q \colon V \to k

is called a quadratic refinement of ,\langle -,-\rangle if

v,w=q(v+w)q(v)q(w)+q(0) \langle v,w\rangle = q(v + w) - q(v) - q(w) + q(0)

for all v,wVv,w \in V.

If such qq is indeed a quadratic form in that q(tv)=t 2q(v)q(t v) = t^2 q(v) then q(0)=0q(0) = 0 and

v,v=2q(v). \langle v , v \rangle = 2 q(v) \,.

This means that a quadratic refinement by a quadratic form always exists when 2k2 \in k is invertible. Otherwise its existence is a non-trivial condition. One way to express quadratic refinements is by characteristic elements of a bilinear form. See there for more.

References

The theory of quadratic forms emerged as a part of (elementary) number theory, dealing with quadratic diophantine equations, initially over the rational integers

The terminology “form” possibly originated with

  • Leonhard Euler, Novae demonstrations circa divisors numerorum formae xx+nyyx x + n y y, Acad. Petrop. recitata, Nov 20, 1775, published poshumously

(which is cited as such in Gauss 1798, paragraph 151).

First classification results for forms over the integers were due to

(which speaks of formas secundi gradus)

  • Herrmann Minkowski, Grundlagen für eine Theorie der quadratischen Formen mit ganzzahligen Koeffizienten, Mémoires présentés par divers savants a l’Acad´emie des Sciences de l’institut national de France, Tome XXIX, No. 2. 1884.

  • Herrmann Minkowski, Untersuchungen über quadratische Formen. Bestimmung der Anzahl verschiedener Formen, die ein gegebenes Genus enthält, Königsberg 1885; Acta Mathematica 7 (1885), 201–258

See also

  • Rudolf Scharlau, Martin Kneser’s work on quadratic forms and algebraic groups, 2007 (pdf)

Course notes include for instance

  • On the relation between quadratic and bilinear forms (pdf)

  • Bilinear and quadratic forms (pdf)

  • section 10 in Analytic theory of modular forms (pdf)

Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See

Revised on March 16, 2015 19:49:44 by Urs Schreiber (88.128.80.39)