《组合学中的多项式方法》第二章习题选做
Chapter 2 Fundamental examples of the polynomial method
\(\newcommand{\LL}{\mathfrak L} \newcommand{\ZZ}{\mathbb Z} \newcommand{\FF}{\mathbb F} \newcommand{\RR}{\mathbb R} \DeclareMathOperator{\poly}{poly}\)
Exercise 2.1. Given a set of \(N\) points in \(\RR^3\), we proved that there is a non-zero polynomial of degree \(\lesssim N^{1/3}\) that vanishes at all the points. Given any \(N\) lines in \(\RR^3\) prove that there is a non-zero polynomial of degree \(\lesssim N^{1/2}\) that vanishes on all the lines.
State and prove a similar result for \(k\)-planes in \(\RR^n\) for any dimensions \(k, n\).
Proof. For each polynomial \(f\in \poly_D(\RR^n)\), for a \(k\)-plane, we can parametrize it by a linear function \(l \colon \RR^k\to \RR^n\), thus \(f(l) \in \poly_D(\RR^k)\). For the lines \(l_1,\dots,l_N\), the mapping \(f \mapsto (f(l_1),\dots,f(l_N))\) maps from a vector space of dimension \(\binom{n+D}{D}\) to a vector space of dimension \(N \binom{k+D}{D}\). When \(\binom{n+D}{D} > N\binom{k+D}{D}\), we have a nontrivial kernel, this happens at \(D \lesssim N^{1/(n-k)}\). \(\square\)
Exercise 2.6. We consider a collection of curves \(\Gamma_a \subset \FF_q^n\) parametrized by \(a\in \FF_q^{n-1}\). For each \(a\in \FF^{n-1}_q\), \(1\leq j\leq n-1\), suppose that \(Q_{a,j} \in \poly_d(\FF_q)\). Let \(\Gamma_a\) be defined as the graph:
\[\Gamma_a := \{ (Q_{a,1}(t), Q_{a,2}(t),\dots, Q_{a,n-1}(t), t)\in \FF_q^{n} \arrowvert t\in \FF_q \}. \]Suppose also that \((Q_{a,1}(0), \dots, Q_{a,n-1}(0)) = a\), so that \((a, 0) \in \Gamma_a\). Prove that there is a constant \(c(d, n) > 0\) so that
\[\left|\bigcup_{a\in \FF_{q}^{n-1}} \Gamma_a \right| \geq c(d,n)q^n. \]
Proof. Suppose a polynomial \(F\) vanishes on \(K\) with \(\deg F < q/d\). Then for each \(a\in \FF_q^{n-1}\), we have the polynomial
vanishes on \(\FF_q\), and \(\deg f_a(t) < q\), we must have \(f_a(t) = 0\). Suppose \(F\) is nonzero, we have
where we have some minimal \(r\) such that \(F_r\neq 0\). Then we have
the coefficient on the \(r\)th order should be \(F_r(a) = 0\), since this holds for all \(a\), we have \(F_r = 0\), contradiction, so such \(F\) must be zero.
Therefore, for \(D<q/d\) we have \(|K| \geq \binom{n+D}{D} \geq (D+1)^n/n!\), thus we have \(|K| \geq (q/d)^n/n!\), we can take \(c(d, n) = 1/n!d^n\). \(\square\)
Exercise 2.6. The joints problem also makes sense in higher dimensions, and [KSS] and [Q] proved a generalization of Theorem 2.12 to all dimensions.
If \(\LL\) is a set of \(\LL\) lines in \(\RR^n\), a joint of \(\LL\) is defined to be a point that lies in \(n\) lines of \(\LL\) pointing in linearly-independent directions.
Theorem 2.17. A set of \(L\) lines in \(\RR^n\) determines at most \(C_n L^{n/(n-1)}\) joints.
Prove this theorem.
Let \(J\) be the set of joints, we know that when \(\binom{D+n}{n} > |J|\), there must be a polynomial \(F\) vanishes on \(J\) such that \(\deg F \leq D\). When \(D > n|J|^{1/n}\), we have \(|J| < n^{-n}D^n \leq \binom{D+n}{n}\), so we must have the minimal \(\deg F \leq n |J|^{1/n}\).
Suppose every line \(\ell\in \LL\) pass through \(> n|J|^{1/n}\) joints, we have \(F(\ell(t))\) vanishes on \(> \deg F\) points, thus \(F(\ell(t)) = 0\) for all \(\ell \in \LL\). Then for any joint \(j\in J\), we have \(n\) linearly independent lines \(\ell\) pass through \(j\) and \(F(\ell) = 0\), we have \(\nabla F\) vanishes on \(J\) in any direction, by the minimality of \(F\), we have \(\nabla F=0\) in any directions, thus \(F\) is a constant, which is not possible.
Therefore, we can conclude that for any line set \(\LL\), it contains a line \(\ell\) that pass through at most \(n|J|^{1/n}\) joints. By repeatedly removing such line, we have \(|J| \leq n L |J|^{1/n}\), thus \(|J| \leq (nL)^{n/(n-1)}\), the constant \(C_n = n^{n/(n-1)}\) is enough. A more precise bound is \(C_n \leq (n!)^{1/(n-1)}\). \(\square\)
