[Paper Review] Combinatorial problems in finite geometry and lacunary polynomials

Aart Blokhuis|ArXiv.org|Apr 28, 2003

Limits and Structures in Graph Theory28 references3 citations

TL;DR

This paper applies Rédei's theory of lacunary polynomials to solve combinatorial problems in finite geometry, particularly concerning blocking sets and direction sets in finite projective planes. It establishes tight bounds on the size of blocking sets and characterizes functions with few directions, proving that minimal blocking sets in $PG(2,p)$ have size at least $3(p+1)/2$, with equality only under specific polynomial conditions.

ABSTRACT

We describe some combinatorial problems in finite projective planes and indicate how Rédei's theory of lacunary polynomials can be applied to them.

Motivation & Objective

• To apply Rédei's theory of lacunary polynomials to solve extremal combinatorial problems in finite projective planes.
• To characterize functions $f: GF(q) \to GF(q)$ that determine few directions, particularly those with $|D_f|$ close to the theoretical minimum.
• To establish tight lower bounds on the size of non-trivial blocking sets in $PG(2,q)$, especially when $q$ is prime.
• To classify configurations of blocking sets achieving minimal size, using algebraic and polynomial methods.
• To extend results on direction sets and blocking sets to higher-fold blocking sets and maximal arcs, particularly in the case of $n$-fold blocking sets with $t < q^{1/6}$.

Proposed method

• Utilizes Rédei's auxiliary polynomial $R(X,Y) = \prod_{w \in GF(q)} (X - wY + f(w))$ to encode the graph of a function and its direction set.
• Analyzes the specialization $R(X,y)$ for $y \in D_f$, showing it becomes a lacunary polynomial $X^q + g(X)$ with $\deg(g) \leq q - |D_f| - 1$, linking direction count to polynomial sparsity.
• Applies the theory of lacunary polynomials, particularly the structure of $f(X) = X^q g(X) + h(X)$ with $\gcd(g,h)=1$, to derive degree constraints on $g$ and $h$.
• Uses the concept of $p^e$-th powers to classify functions based on their algebraic structure, especially when $f \in GF(q)[X^{p^e}]$.
• Applies the Rédei polynomial to blocking sets by coordinatizing the plane so that the line at infinity is tangent, leading to a lacunary form $f(X) = X^q g(X) + h(X)$ with $\deg(g) = d$ and $\deg(h) \leq d+1$.
• Employs symmetric function theory and properties of elementary symmetric polynomials to deduce that $r_i(Y) \equiv 0$ for $i > |D_f|$, leading to lacunary structure in the polynomial expansion.

Experimental results

Research questions

• RQ1 What is the minimal size of a non-trivial blocking set in $PG(2,p)$ for prime $p$, and what configurations achieve this bound?
• RQ2 Which functions $f: GF(q) \to GF(q)$ determine only a small number of directions, and how can they be algebraically characterized?
• RQ3 Under what conditions does a set of $q + d$ affine points in $PG(2,q)$, together with $d$ points at infinity, form a blocking set, and what constraints does this impose on the associated polynomial?
• RQ4 Can the structure of maximal arcs in $PG(2,q)$ be fully characterized using lacunary polynomial methods, especially for odd $n$?
• RQ5 What are the necessary and sufficient conditions for a function $f$ to have $|D_f| < (q+3)/2$, and how does this relate to its algebraic form over subfields?

Key findings

• For $q = p$ prime, any non-trivial blocking set in $PG(2,p)$ has size at least $3(p+1)/2$, and equality holds only if each point lies on exactly $(p-1)/2$ tangents.
• If $f: GF(q) \to GF(q)$ is nonlinear and $|D_f| < (q+3)/2$, then $f$ is a linear map over a subfield $GF(p^e)$ with $p^e > 3$ or $p^e = 3$ and $|D_f| = q/3 + 1$.
• For $t$-fold blocking sets in $PG(2,q)$ with $t < q^{1/6}$, the minimal size is $t(q + \sqrt{q} + 1)$, achieved if and only if the set is a union of $t$ disjoint Baer subplanes.
• In the case $q = p$ prime, the minimal blocking set of size $3(p+1)/2$ is unique up to equivalence for $p \neq 7,13$, and all such configurations can be classified via computer search for $p \leq 37$.
• The Rédei polynomial $R(X,Y)$ specializes to a lacunary polynomial $X^q + g(X)$ when $Y = y \in D_f$, and the degree of $g$ gives a lower bound on $|D_f|$.
• For $f(X) = X^q g(X) + h(X)$ with $\gcd(g,h)=1$, the degree $k = \max(\deg g, \deg h)$ satisfies a series of lower bounds depending on the maximal $e$ such that $f \in GF(q)[X^{p^e}]$, with tight constraints in each case.