Quaternary Quadratic Forms

Question

What is a Quaternary Quadratic Form? I've looked for a definition online and cannot find a precise clear definition.

I am not taking a course. Just reading about quadratic forms.

Thank you.

Answer 1

A quaternary quadratic form is a quadratic form in 4 variables. (A binary form is one in 2 variables, a ternary in 3, etc.)

Answer 2

For positive ones see

http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/nipp.html

http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/nipp.html#form

http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/d4to457.html

I do not know of any tables of indefinite quaternaries. The interesting examples are of this type: $$ w^2 - 2 x^2 + 3 y^2 - 6 z^2. $$ For real irrational values of the variables, it is easy to make this add up to $0,$ just take $w=\sqrt 2, x=1, y=0,z=0.$ For integers (and so rationals) the sum cannot be $0$ unless all four of $w,x,y,z=0.$ This is simply because, for integers, $w^2 - 2 x^2 + 3 y^2 - 6 z^2 \equiv 0 \pmod 9$ means $w,x,y,z \equiv 0 \pmod 3. $ If there were any nontrivial integer expression $w^2 - 2 x^2 + 3 y^2 - 6 z^2=0,$ there would also be one with $\gcd(w,x,y,z) = 1;$ but we see that this is not possible.