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Let $p(x,y,z): \mathbb{F}^3 \to \mathbb{F}$ be a trivariate polynomial of degree $d \ll |\mathbb{F}|$. We choose uniformly at random an affine $1$-dimentional space $\ell = \{(a_1,a_2,a_3)t + (b_1,b_2,b_3) : t \in \mathbb{F}\}$. What is a good bound on the probability that the restriction of $p$ to $\ell$ (as a univariate polynomial) is of degree strictly less than $d$?

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Let's say $q(x,y,z)$ is the homogeneous part of $p$ of degree $d$. The degree $d$ part of the restriction of $p$ to $l$ (as a polynomial in $t$), is $q(a_1 t, a_2 t, a_3 t) = q(a_1, a_2, a_3) t^d$. Therefore the question reduces to the probability that a given (non-zero) homogenous polynomial of degree $d$ vanishes on a uniformly chosen element of $\mathbb{F}^3$. –  Magdiragdag Aug 16 '13 at 8:53
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