Chapter 8 : Distinct intersection points

Claim 1.1. Let L be a set of n lines in R. Then there exists a nontrivial polynomial f ∈ R[x1, x2, x3] of degree smaller than 3 √ n that vanishes on all the lines of L. Proof. Let P be a set of at most 4n points, that is obtained by arbitrarily choosing 4 √ n points from every line of L. Since ( 3 √ n+3 3 ) > 4n, by Lemma 2.1 of Chapter 5 there exists a nontrivial polynomial f ∈ R[x1, x2, x3] of degree at most 3 √ n that vanishes on P . Consider a line l ∈ L. Since f vanishes on at least 4 √ n points of l, Bézout’s theorem implies that l is contained in Z(f).