The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where... more
The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\binom {n,q}{cn}-\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor$ or $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\lfloor\log_2{n}\rfloor$ or $q=\lfloor\log_2{n}\rfloor+1$.
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We obtain asymptotic formulas for sums of coefficients over arithmetic progressions of polynomials related to the Borwein conjectures. Let $a_i$ denote the coefficient of $q^i$ in the polynomial... more
We obtain asymptotic formulas for sums of coefficients over arithmetic progressions of polynomials related to the Borwein conjectures. Let $a_i$ denote the coefficient of $q^i$ in the polynomial $\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s$, where $p$ is an odd prime, and $n, s$ are positive integers. In this note, we prove that $$\Big|\sum_{i=b\ \text{mod}\ 2pn}a_i-\frac{(p-1)p^{sn-1}}{2n}\Big|\leq p^{sn/2},$$ if $b$ is divisible by $p$, and $$\Big|\sum_{i=b\ \text{mod}\ 2pn}a_i+\frac{p^{sn-1}}{2n}\Big|\leq p^{sn/2},$$ if $b$ is not divisible by $p$. This improves a recent result of Goswami and Pantangi.
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A conjecture of Borwein asserts that for any positive integers $n$ and $k$, the coefficient $a_{3k}$ of $q^{3k}$ in the expansion of $\prod_{j=0}^n (1-q^{3j+1})(1-q^{3j+2})$ is nonnegative. In this paper we prove that for any $0 \leq... more
A conjecture of Borwein asserts that for any positive integers $n$ and $k$, the coefficient $a_{3k}$ of $q^{3k}$ in the expansion of $\prod_{j=0}^n (1-q^{3j+1})(1-q^{3j+2})$ is nonnegative. In this paper we prove that for any $0 \leq k\leq n$, there is a constant $0 0.$$
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This exploratory note is an introduction to additive combinatorics, with an emphasis on sum-product phenomenon and some arithmetical applications. Most results and proofs can be found in [3], [5], [44], [62].
The multinomial coefficient ( n,q k ) is defined to be the coefficient of xk in (1 + x + x2 + · · · + xq−1)n. It is conjectured that for given n > 2, T (n, q) := ( n,q cn ) − ( n,q−1 cn ) is unimodal and the maximum occurs at q =... more
The multinomial coefficient ( n,q k ) is defined to be the coefficient of xk in (1 + x + x2 + · · · + xq−1)n. It is conjectured that for given n > 2, T (n, q) := ( n,q cn ) − ( n,q−1 cn ) is unimodal and the maximum occurs at q = ⌊log1+ 1 c n⌋ or q = ⌊log1+ 1 c n⌋ + 1. As an attempt to prove this conjecture, we give an asymptotic estimate for ( n,q cn ) as n tends to infinity, where c is a positive integer.
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In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. If... more
In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. If $q^h-1$ has a prime divisor $r$ with $r=O((h\log q)^c)$, then there is a constant $0 0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+\epsilon'}(q^h))$ containing a non-quadratic element in $\mathbb{F}_{q^h}$. On the other hand, the choice of $h=2$ shows that for any odd prime power $q$, there is an explicit subset of cardinality $O(\sqrt{q})$ containing a non-quadratic element in $\mathbb{F}_{q^2}$, essentially improving a $O(q)$ construction by Coulter and Kosick \cite{CK}. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided $\phi(q^h-1)$ is very small, where $\phi$ is the Euler's totient function.
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In this paper, we obtain an asymptotic formula for the number of codewords with a fixed distance to a given received word of degree $k+m$ in the standard Reed-Solomon code $[q, k, q-k+1]_q$. Previously, explicit formulas were known only... more
In this paper, we obtain an asymptotic formula for the number of codewords with a fixed distance to a given received word of degree $k+m$ in the standard Reed-Solomon code $[q, k, q-k+1]_q$. Previously, explicit formulas were known only for the cases $m=0, 1, 2$.
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Let $\mathbb{F}_q$ be the finite field of $q$ elements and $a_1,a_2, \ldots, a_k, b\in \mathbb{F}_q$. We investigate $N_{\mathbb{F}_q}(a_1, a_2, \ldots,a_k;b)$, the number of ordered solutions $(x_1, x_2, \ldots,x_k)\in\mathbb{F}_q^k$ of... more
Let $\mathbb{F}_q$ be the finite field of $q$ elements and $a_1,a_2, \ldots, a_k, b\in \mathbb{F}_q$. We investigate $N_{\mathbb{F}_q}(a_1, a_2, \ldots,a_k;b)$, the number of ordered solutions $(x_1, x_2, \ldots,x_k)\in\mathbb{F}_q^k$ of the linear equation $$ a_1x_1+a_2x_2+\cdots+a_kx_k=b$$ with all $x_i$ distinct. We obtain an explicit formula for $N_{\mathbb{F}_q}(a_1,a_2, \ldots, a_k;b)$ involving combinatorial numbers depending on $a_i$'s. In particular, we obtain closed formulas for two special cases. One is that $a_i, 1\leq i\leq k$ take at most three distinct values and the other is that $\sum_{i=1}^ka_i=0$ and $\sum_{i\in I}a_i\neq 0$ for any $I\subsetneq [k]$. The same technique works when $\mathbb{F}_q$ is replaced by $\mathbb{Z}_n$, the ring of integers modulo $n$. In particular, we give a new proof for the main result given by Bibak, Kapron and Srinivasan, which generalizes a theorem of Schonemann via a graph theoretic method.