= Polynomial restriction to a line
{title2=$R_\ell f(t)=f(\mathbf a+t\mathbf v)$}
For a line $\ell=\{\mathbf a+t\mathbf v:t\in\mathbb R\}$ with $\mathbf v\ne0$, restricting a degree-at-most-$d$ <multivariate polynomial> gives a univariate polynomial in $t$ of degree at most $d$. Its $d+1$ coefficients are <linear functionals> of the original coefficient vector. Vanishing on the whole line is therefore equivalent to at most $d+1$ homogeneous linear conditions. Alternatively, $d+1$ distinct roots force the restriction to be identically zero.
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