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

$P(v)=\sum _{k=0}^{p}A_{k}(v,\dots ,v)$

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

$P(v_{1},\dots ,v_{n})=\sum _{k=0}^{p}\sum _{j=0}^{m_{k}}A_{1,j,k}(v_{1},\dots ,v_{1})\dots A_{n,j,k}(v_{n},\dots ,v_{n})$

where Ai,j,k ∈ Lpi,j,ksym  with  $\sum _{i=0}^{n}p_{i,j,k}=k$. In this case P is called homogeneous if we only have the k = p  summand in the above expression.

References

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