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Register a new userOn the unique representability of spikes over prime fields
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For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $kin{1, 2, ..., n - 1}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $n ge 2p-1$. Moreover, $M$ is uniquely representable over $GF(p)$.
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In this paper we obtain a new lower bound on the ErdH{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $Asubset mathbb{F}_p^2$ with $|A|le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |Delta(A)| gg |A|^{frac{1}{2}+frac{149}{4214}}.$$ Our result gives a new lower bound of $|Delta{(A)}|$ in the range $|A|le p^{1+frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in $mathbb{F}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw.
Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ over $F$ have rank $leq r$ ? This question is classical, and the answer ($q^{2r}$ when $rleqminleft{ p,qright} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},ldots,x_{k-1}$ for some $kleq rleqminleft{ p,qright} $, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},ldots,x_{p+q}$ such that the resulting Hankel matrix $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ has rank $leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.
Let $mathbb{F}_p$ be a prime field, and ${mathcal E}$ a set in $mathbb{F}_p^2$. Let $Delta({mathcal E})={||x-y||: x,y in {mathcal E} }$, the distance set of ${mathcal E}$. In this paper, we provide a quantitative connection between the distance set $Delta({mathcal E})$ and the set of rectangles determined by points in ${mathcal E}$. As a consequence, we obtain a new lower bound on the size of $Delta({mathcal E})$ when ${mathcal E}$ is not too large, improving a previous estimate due to Lund and Petridis and establishing an approach that should lead to significant further improvements.
Given $E subseteq mathbb{F}_q^d times mathbb{F}_q^d$, with the finite field $mathbb{F}_q$ of order $q$ and the integer $d ge 2$, we define the two-parameter distance set as $Delta_{d, d}(E)=left{left(|x_1-y_1|, |x_2-y_2|right) : (x_1,x_2), (y_1,y_2) in E right}$. Birklbauer and Iosevich (2017) proved that if $|E| gg q^{frac{3d+1}{2}}$, then $ |Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| gg q^{frac{10}{3}}$, then $ |Delta_{2, 2}(E)| gg q^2$. In this paper, we present extensions and improvements of these results.
We establish an uncertainty principle for functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ with constant support (where $p mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ for which $|text{supp}; {f}| = S$ must satisfy $|text{supp}; hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredis theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.
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