nLab quadratic function

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 $H$, since we can regard $G$ and $H$ as $\mathbb{Z}$- modules, the above definition specializes to this particular case. 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 $r \in \mathbb{Z}$,
  $$q(r x) = \sum_{i=1}^{r^2} q(x)$$

See also

• completing the square

• quadratic formula

• quadratic form, quadratic refinement

• linear function, polynomial function

• real quadratic function

• quadratic abelian group

• cubic function

References

See also:

• Wikipedia, Quadratic function

Discussion of quadratic functions in the form of quadratic refinements of intersection pairings (in cohomology), as a phenomenon in algebraic topology, differential topology as well as in string theory:

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory, J. Diff. Geom. 70 (2005) 329-452 $[$arXiv:math/0211216, euclid.jdg/1143642908$]$