Restriction¶
If polynomial \(q\) is a restriction of a polynomial \(f\) to a line \(l\), then for every random point \(r\), we have \(q(r) = f(l(r))\).
Example:
\(l(t) = (t + 2,4 - 2t)\)\ \(f(x, y) = 3xy + 2y\)\ \(q = 3(t+2)(4-2t) + 2(4-2t) = -6t^2 - 4t + 32\)
Random point: \(r = 5 \rightarrow l(r) = (7, -6)\)
\(q(r)=q(5)\) \(\iff q(r)=(-6) * 5^2-4 * 5+32 = -138\)
\(f(l(r))= f(l(5)) = f(7,-6)\) \(\iff f(l(r)) =3 * 7 * (-6)+2 * (-6)=-138\)