The exact definition of a form

Question

I am studying Fulton's Algebraic Geometry book and I can't understand what a form is! Sometimes it factors a polynomial $F$ to something like: $$F=f_0 + f_1 + f_2 +...+ f_m$$ and it calls $f_i$ a "form". What is the exact definition of a form?

Answer 1

Putting together the answers in the comments to the question:

Viktor Vaughn writes: This is defined on the very first page of the book: "We call F homogeneous, or a form, of degree $d$, if all coefficients $a_{(i)}$ are zero except for monomials of degree $d$."

Quasi writes: Thus, for example, $x^2+3xy−2yz$ is a form of degree $2$, since all terms have total degree $2$, but $x^2+3x^2y−2yz$ is not a form.

(Note that $3x^2y$ has degree $2+1=3$.)

See also https://en.wikipedia.org/wiki/Homogeneous_polynomial.