# 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

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

for v ∈ V and Ak ∈ Lksym = the set of all F-valued symmetric k-linear forms for k = 0, ..., pP  is called homogeneous of degree p  if P = Ap  above.

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

$P(v_1,\dots,v_n)=\sum^{p}_{k=0}\sum^{m_k}_{j=0}A_{1,j,k}(v_1,\dots,v_1)\dots A_{n,j,k}(v_n,\dots,v_n)$

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