In the litterature we see the terminology "multilinear form" or "$n$-form". I'm used to refer the word "form" to mean a homogeneous polynomial. but here we define it as a map $f:V^n\to F$, ($V$ is an $F$-vector space), such that $f$ is linear on each of its components. I'm confused by terminology, for example is there some connection between a bilinear form and a quadratic form?
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If $B : V^2 \to F$ is a symmetric bilinear form, then $B(v, v) : V \to F$ is a quadratic form. If $F$ does not have characteristic $2$, then conversely if $q(v) : V \to F$ is a quadratic form, then $$B(v, w) = \frac{q(v + w) - q(v) - q(w)}{2}$$ is a symmetric bilinear form. More generally, if $f : V^n \to F$ is a symmetric multilinear form, then (in sufficiently large characteristic) it can be (not quite canonically) identified with an element of the symmetric power $\text{Sym}^n(V^{\ast})$. This symmetric power lies in the symmetric algebra $\text{Sym}(V^{\ast})$, which one can think of as an abstract form of a polynomial ring (it is the ring of polynomial functions on $V$). |
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