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This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not even so much interested in characterizing the solutions as I am in counting them (or showing some relation between them), but any help would be greatly appreciated (as well as a pointer to a good introductory article/text on quadratic and modular forms in general).


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You call n variables x_i and m variables y_j. There's no need for that: just use one set of variables x_1,...,x_n. A quadratic form in n variables is a homogeneous polynomial of degree 2: Q = sum_{i <= j} a_{ij}x_ix_j. As long as p is not 2, a nondegenerate quadratic form over F_p can be written in diagonal form as x_1^2 + ... + x_{n-1}^2 + dx_n^2, where the nonzero number d only matters modulo squares (so basically there are just two choices). For degenerate quadratic forms in n variables you just use fewer variables in the diagonalized form (like Q = x^2 as a function of x and y). – KConrad Aug 21 '10 at 5:17
See Ireland-Rosen Section 7 Chapter 8 for a general formula for the number of solutions to a diagonal equation over a finite field, which includes your question as a special case. (You don't need modular forms for this, except in the archaic sense that "modular form" used to literally mean a homogeneous polynomial over a "modular", i.e., finite, field. This was used in some paper of Dickson, but the term modular form no longer has that meaning.) – KConrad Aug 21 '10 at 5:19
See also Theorem 2 Chapter 10 Section 3 of Ireland--Rosen (counting solutions projectively, which is reasonable for a homogeneous polynomial). – KConrad Aug 21 '10 at 5:21
Deifnitely not a silly question, by the way. – Pete L. Clark Aug 21 '10 at 11:03

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