What is the definition of a non-degenerate homogeneous quadratic form over a finite field?

Question

I read in some finite geometry notes by S. Ball and Z. Weiner the following:

A conic is a set of points of $PG(2,q)$ that are zeros of a non-degenerate homogeneous quadratic form (in $3$ variables), for example, $f(\mathbb{x})= x^2-yz$.

I try to find a definition for non-degenerate but I can't find anything for finite fields, except for non-degenerate bilinear forms. Here they give a definition of a non-degenerate quadratic form for two variables.

What is the general definition of a non-degenerate homogeneous quadratic form in 3 variables over a finite field?

Answer 1

A quadratic form $f$ is non-degenerate if there is a basis $(v_1,v_2,\ldots, v_n)$ such that $$ f(x_1v_1 + x_2v_2 + \cdots + x_nv_n) = \lambda_1x_1^2 + \lambda_2x_2^2+\cdots+\lambda_nx_n^2 $$

for some constants $\lambda_i$ and all $x_i$, such that the $\lambda_i$ are all nonzero.

Finding this basis (and the associated $\lambda_i$s) corresponds to diagonalizing the symmetric matrix representation of the quadratic form.

(Characteristic 2 is probably a special case, though, since there are quadratic forms in characteristic 2 that are not represented by any symmetric matrix -- such as $x^2-yz$!).

Answer 2

A very detailed discussion about this, that also includes many equivalent definitions, can be found in this note. As Henning pointed out, when the finite field has characteristic $2$, things get trickier and indeed the definitions in this note is only for finite fields of odd characteristic.