On L-functions of certain exponential sums

Let ${\mathbb{F}}_q$ denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials$f(x)=a_1{x}_{n+1}(x_1+\frac{1}{x_1})+\cdots +a_n{x}_{n+1}(x_n+\frac{1}{x_n})+a_{n+1}{x}_{n+1}+\frac{1}{x_{n+1}}$ where $a_i\in {\mathbb{F}}_q^{⁎}$, $i=1,2,\dots ,n+1$. When $n=2$, the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group ${\Gamma }_0(p)$, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., $n⩽3$.     

The expansion of a chord diagram and the Tutte polynomial

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ is called nonintersecting if $E$ contains no crossing. For a chord diagram $E$ having a crossing $S=\{x_1{x}_3,x_2{x}_4\}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1=(E?S)\cup \{x_2{x}_3,x_4{x}_1\}$ or $E_2=(E?S)\cup \{x_1{x}_2,x_3{x}_4\}$. For a chord diagram $E$, let $f\left(E\right)$ be the chord expansion number of $E$, which is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from $E$ with a finite sequence of expansions.In this paper, it is shown that the chord expansion number $f\left(E\right)$ equals the value of the Tutte polynomial at the point $(2,?1)$ for the interlace graph $G_E$ corresponding to $E$. The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) [13].     

On tiling the integers with 4-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1,\dots ,x_n\}$ of integers where $x_1<?<x_n$, let the gap sequence of this set be the unordered multiset $\{d_1,\dots ,d_{n?1}\}=\{x_{i+1}?x_i:i\in \{1,\dots ,n?1\}\}$. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\ge r_0$, the set of integers can be partitioned into 4-sets with gap sequence $p,q$, $r$.     

Subspace partitions of Fqn containing direct sums

Let $q$ be a prime power and $n$ be a positive integer. A subspace partition of $V={\mathbb{F}}_q^n$, the vector space of dimension $n$ over ${\mathbb{F}}_q$, is a collection $\Pi$ of subspaces of $V$ such that each nonzero vector of $V$ is contained in exactly one subspace in $\Pi$; the multiset of dimensions of subspaces in $\Pi$ is then called a Gaussian partition of $V$. We say that $\Pi$contains a direct sum if there exist subspaces $W_1,\dots ,W_k\in \Pi$ such that $W_1\oplus ?\oplus W_k=V$. In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers $a_1$ and $a_2$ with $n>a_1>a_2\ge 1$, our main theorem shows that if $\Pi$ is a subspace partition of ${\mathbb{F}}_q^n$ with $m_i$ subspaces of dimension $a_i$ for $i=1,2$, then $\Pi$ contains a direct sum when $a_1{x}_1+a_2{x}_2=n$ has a solution $(x_1,x_2)$ for some integers $x_1,x_2\ge 0$ and $m_2$ belongs to the union $I$ of two natural intervals. The lower bound of $I$ captures all subspace partitions with dimensions in $\{a_1,a_2\}$ that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when $m_2?I$ or when the condition on the existence of a nonnegative integral solution $(x_1,x_2)$ is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when $q\to 1$) as well as subspace and set partitions.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.