TITLE: What, if anything, makes homogeneous polynomials so great?
QUESTION [21 upvotes]: It should be obvious from the question that I am not any kind of algebraic geometer, so if there are definitions of hom-polys as comonoidal dyadic functors or whatnot, let's leave that to one side for the purposes of this question. I really mean hom-polys in the most pedestrian sense possible.
From the outside, it seems that homogeneous polynomials get a lot of attention in alg-geom. (Perhaps in other areas as well?) I know that they are, well, homogeneous with respect to dilations, and that this allows one to look at their zeros in projective space in a natural way.
I usually like to keep a safe distance between myself and projective space, and have always looked at hom-polys as "merely" a technical tool. But, just the other day, I was able to quickly solve a small, elementary number-theoretic problem by converting it into "homogeneous" form. (Some vague memories of homogeneous problem-solving heuristics prompted this.) To be precise, the problem was that of computing how many solutions there are to m^2 + n^2 = 1 in Z_p, where p is a prime congruent to 3 mod 4. (Of course this is trivial by a change of variables when p = 1 mod 4.) The problem seemed unfriendly at first, but passing to the related question of the solutions to m^2 + n^2 - t^2 = 0 revealed a lot of symmetry and made it quite trivial. (Of course, solving small cases of the first problem showed a lot of symmetry, but it wasn't obvious how to get a handle on it.)
My question is: do hom-polys help to solve a lot of seemingly unrelated problems via some process of homogeneization of the problem? Do you have examples? Is this one reason for their popularity? I'm looking for heuristics mostly, but if you think an actual theorem makes precise or helps formulate a heuristic, go nuts.

REPLY [3 votes]: As you already noticed,  thinking in terms of homogeneous coordinates in a complex projective space $\mathbb{P}^n$, in order to define a function (any old complex-valued function, not necessarily holomorphic or even continuous) on $\mathbb{P}^n$ we must require that $f$ is homogeneous of order zero, i.e., $f(\lambda x)=f(x)$ for any $\lambda \in \mathbb{C}\setminus 0$ and any $x \in \mathbb{P}^n$. (I am cutting some corners with this notation.) But in order to define a zero set of a function on   $\mathbb{P}^n$ , we have a little more leeway: requiring that the the ``function" be homogeneous of order $k \geq 0$, i.e.,  $f(\lambda x)=\lambda^kf(x)$ for any $\lambda \in \mathbb{C}\setminus  0$ and any $x \in \mathbb{P}^n$, will do. A more profound statement (=Chow's theorem) is true (sort of converse to the above): every analytic set in $\mathbb{P}^n$ is the (common) zero set of finitely many homogeneous polynomials. 
Aside 1: This point of view generalizes to toric varieties, since they admit an analog of homogeneous coordinates (first introduced in the smooth case in 
MR1106194 Audin, Michèle: The topology of torus actions on symplectic manifolds. Translated from the French by the author. Progress in Mathematics, 93. Birkhäuser Verlag, Basel, 1991. 181 pp. ISBN: 3-7643-2602-6, then developed for the general case in 
MR1299003 Cox, David A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4 (1995), no. 1, 17–50)
Aside 2:  the interpretation of sections of a line bundle over a complex manifold as homogeneous functions has nothing to do with projectivity and/or Serre's theorem (as an analyst, I'll stick to the language of complex manifolds and line bundles instead of schemes and sheaves). It is more general and follows  from the relation between the $k$-th tensor power of the line bundle $\pi: L \to X$ over an $n$-dimensional complex manifold $X$ and the dual bundle $\pi': L' \to X$. Namely (as can be checked using respective definitions, taking into account relations between trivializations), every holomorphic function $f$ on $L'$ which is homogeneous of degree $k$ on the fibers is a section in $L^k$ (not just a ''stand-in").  In the specific case of $X=\mathbb{P}^n$ and $L=\mathcal{O}(1)$, the space of sections $\Gamma(\mathbb{P}^n, \mathcal{O}(k))$ is isomorphic to the space of homogeneous polynomials of degree $k$ in the variables $z_0,...,z_n$.