Polynomials on vector spaces

In mathematics, if F is a field, a single-variable F-valued polynomial of degree ≤ p on a vector space V is a map P : V → F of the form

for v ∈ V and A_k ∈ L^k_sym = the set of all F-valued symmetric k-linear forms for k = 0, ..., p.  P  is called homogeneous of degree p  if P = A_p  above.

Similarly, one can define an n-variable F-valued polynomial of degree  ≤ p  on V  to be

where Ai,j,k ∈ Lpi,j,ksym  with  .  In this case P  is called homogeneous if we only have the k = p  summand in the above expression.

See also

• Polynomial
• Chern-Weil homomorphism

References

• Ralph Abraham, Joel Robbin. Transversal Mapppings and Flows, p 7. W. A. Benjamin Inc. 1967.