Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

### In commutative rings

For a commutative ring $R$, a quadratic function is a function $f \colon R \to R$ with elements $a \in R$, $b \in R$, $c \in R$ such that for all $x \in R$,

$f(x) = a \cdot x^2 + b \cdot x + c$

where $x^2$ is the canonical square function of the multiplicative monoid.

### Between $R$-modules

Given a commutative ring $R$ and $R$-modules $M$ and $N$, an $R$-quadratic function on $M$ with values in $N$ is a map $q: M \to N$ such that the following properties hold:

• (cube relation) For any $x,y,z \in M$,
$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0$
• (homogeneous of degree 2) For any $x \in M$ and any $r \in R$,
$q(r x) = r^2 q(x)$

### Between abelian groups

Given abelian groups $G$ and $H$, a quadratic function on $G$ with values in $H$ is a map $q: G \to H$ such that the following properties hold:

• (cube relation) For any $x,y,z \in G$,
$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0$
• (homogeneous of degree 2) For any $x \in G$ and any $r \in \mathbb{Z}$,
$q(r x) = \sum_{i=0}^{r^2} q(x)$