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Suppose $f(x,y) \in \mathbb{Z}[x,y]$ is a binary form (that is, homogeneous polynomial in two variables). Further suppose that $f$ is irreducible over $\mathbb{Z}$ and has no fixed prime divisor. Let $m \geq 1$ be a fixed modulus. Then the set of solutions to the congruence

$$\displaystyle f(x,y) \equiv 0 \pmod{m}$$

satisfying $x \equiv \omega y \pmod{m}$ forms a lattice. This in turn allows us to use tools from geometry of numbers to study the problem.

However, the same cannot be said when we look at the congruence

$$\displaystyle f(x,y) \equiv c \pmod{m}$$

when $c \ne 0$.

My questions on this matter are as follows:

1) For small $c$, say square-free, can one expect the number of solutions of $f(x,y) \equiv c \pmod{m}$ with $x,y \in [-B,B]$ to be comparable to the number of solutions to $f(x,y) \equiv 0 \pmod{m}$? Why or why not?

2) What can we say about the size of the set of points $(x,y)$ with $x, y \in [-B, B]$ satisfying for at least one prime $B \ll p \ll B^{2-\delta}$ the congruence $f(x,y) \equiv c \pmod{p^k}$ for some fixed positive integer $k \geq 2$ and some small positive constant $\delta > 0$?

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  • $\begingroup$ What is $\omega$? $\endgroup$ – M.D. Jan 2 '14 at 16:10
  • $\begingroup$ $\omega$ is just a fixed element of $\mathbb{Z}/m\mathbb{Z}$ $\endgroup$ – Stanley Yao Xiao Jan 2 '14 at 16:43

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