Contents

# Contents

## Definition

For $V$ a vector space or more generally a $k$-module, then a quadratic form on $V$ is a function

$q\colon V \to k$

which is homogeneous of degree 2 in that for all $v \in V$, $t \in k$

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

and such that the polarization of $q$

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

is a bilinear form.

Written entirely in terms of $q$, the axioms for a quadratic form are:

• $q(t v) = t^2 q(v)$,
• $q(t v + w) + t q(v) + t q(w) = t q(v + w) + t^2 q(v) + q(w)$,
• $q(u + v + w) + q(u) + q(v) + q(w) = q(u + v) + q(u + w) + q(v + w)$.

(Besides the homogeneity, these come from two requirements of a bilinear form to preserve scalar multiplication and addition, respectively.) So we may alternatively define a quadratic form to be a map $q\colon V \to k$ satisfying these three axioms.

A more general quadratic map (or homogeneous quadratic map to be specific) between vector spaces $V$ and $W$ is a map $q\colon V \to W$ that satisfies the above three conditions. (Then an affine quadratic map is the sum of a homogeneous quadratic map, a linear map, and a constant, just as an affine linear map is the sum of a linear map and a constant.)

From the converse point of view, $q$ is a quadratic refinement of the bilinear form $(-,-)$. (This always exists uniquely if $2 \in k$ is invertible, but in general the question involves the characteristic elements of $(-u,-)$. See there for more.)

Quadratic forms with values in the real numbers $k = \mathbb{R}$ are called positive definite or negative definite if $q(v) \gt 0$ or $q(v) \lt 0$, respectively, for all $v \neq 0$. See definiteness for more options.

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 $x x + n y y$, 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)

• Hermann 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.

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

• 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
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