nLab
quadratic function
Redirected from "quadratic functions".
Contents
Context
Algebra
algebra , higher algebra
universal algebra
monoid , semigroup , quasigroup
nonassociative algebra
associative unital algebra
commutative algebra
Lie algebra , Jordan algebra
Leibniz algebra , pre-Lie algebra
Poisson algebra , Frobenius algebra
lattice , frame , quantale
Boolean ring , Heyting algebra
commutator , center
monad , comonad
distributive law
Group theory
Ring theory
Module theory
Contents
Definition
In commutative rings
For a commutative ring R R , a quadratic function is a function f : R → R f \colon R \to R with elements a ∈ R a \in R , b ∈ R b \in R , c ∈ R c \in R such that for all x ∈ R x \in R ,
f ( x ) = a ⋅ x 2 + b ⋅ x + c f(x) = a \cdot x^2 + b \cdot x + c
where x 2 x^2 is the canonical square function of the multiplicative monoid .
Between R R -modules
Given a commutative ring R R and R R - modules M M and N N , an R R -quadratic function on M M with values in N N is a map q : M → N q: M \to N such that the following properties hold:
(cube relation) For any x , y , z ∈ M 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 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 ∈ M x \in M and any r ∈ R r \in R ,q ( r x ) = r 2 q ( x ) q(r x) = r^2 q(x)
Between abelian groups
Given abelian groups G G and H H , since we can regard G G and H H as ℤ \mathbb{Z} - modules , the above definition specializes to this particular case. A quadratic function on G G with values in H H is a map q : G → H q: G \to H such that the following properties hold:
(cube relation) For any x , y , z ∈ G 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 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 ∈ G x \in G and any r ∈ ℤ r \in \mathbb{Z} ,q ( r x ) = ∑ i = 1 r 2 q ( x ) q(r x) = \sum_{i=1}^{r^2} q(x)
See also
References
See also:
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 :
Last revised on August 21, 2024 at 01:49:00.
See the history of this page for a list of all contributions to it.