nLab quadratic function

Contents

Contents

Definition

In commutative rings

For a commutative ring RR, a quadratic function is a function f:RRf \colon R \to R with elements aRa \in R, bRb \in R, cRc \in R such that for all xRx \in R,

f(x)=ax 2+bx+cf(x) = a \cdot x^2 + b \cdot x + c

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

Between RR-modules

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

  • (cube relation) For any x,y,zMx,y,z \in M,
    q(x+y+z)q(x+y)q(x+z)q(y+z)+q(x)+q(y)+q(z)=0q(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 xMx \in M and any rRr \in R,
    q(rx)=r 2q(x)q(r x) = r^2 q(x)

Between abelian groups

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

  • (cube relation) For any x,y,zGx,y,z \in G,
    q(x+y+z)q(x+y)q(x+z)q(y+z)+q(x)+q(y)+q(z)=0q(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 xGx \in G and any rr \in \mathbb{Z},
    q(rx)= i=0 r 2q(x)q(r x) = \sum_{i=0}^{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 May 17, 2022 at 01:14:54. See the history of this page for a list of all contributions to it.