Searcharxiv⌕ Search

arXiv subjects

Daqing Wan

Publications and source records attributed to Daqing Wan.

At least 55 records · Page 3Linked to original sources
arXiv2016

Slopes of eigencurves over boundary disks

Let $p$ be a prime number. We study the slopes of $U_p$-eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra. We give a sharp lower bound of the corresponding Newton polygon. The computation happens over a definite quaternion algebra by Jacquet-Langlands correspondence; it generalizes a prior work of Daniel Jacobs who treated the case of $p=3$ with a particular level. In case when the modular forms have a finite character of conductor highly divisible by $p$, we improve the lower bound to show that the slopes of $U_p$-eigenvalues grow roughly like arithmetic progressions as the weight $k$ increases. This is the first very positive evidence for Buzzard-Kilford's conjecture on the behavior of the eigencurve near the boundary of the weight space, that is proved for arbitrary $p$ and general level. We give the exact formula of a fraction of the slope sequence.

math.NT↗
arXiv2016

Sparse Univariate Polynomials with Many Roots Over Finite Fields

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2(q-1)^{\frac{t-2}{t-1}}$ on the number of cosets in $\mathbb{F}^*_q$ needed to cover the roots of $f$ in $\mathbb{F}^*_q$. Here, we give explicit $f$ with root structure approaching this bound: For $q$ a $(t-1)$-st power of a prime we give an explicit $t$-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of $\mathbb{F}^*_q$. Over prime fields $\mathbb{F}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $Ω\left(\frac{\log p}{\log \log p}\right)$ distinct roots in $\mathbb{F}_p$.

math.NT↗
arXiv2016

On the arithmetic of Z_p-extensions

This paper contains three parts. In the first part, we give a thorough overview of the theory of Artin-Schreier-Witt extensions: this theory allows one to understand the $\mathbf{Z}/p^n\mathbf{Z}$-extensions of any field $K$ of characteristic $p$ via $p$-typical Witt vectors. Let $W_n(K)$ be the ring of $p$-typical Witt vectors of $K$ of length $n$ and let $\wp = F-\mathrm{id}: W_n(K)\longrightarrow W_n(K)$, where $F$ is the Frobenius map and $\mathrm{id}$ is the identity map. Artin-Schreier-Witt theory tells us that the abelian group $W_n(K)/\wp W_n(K)$ represents the set of $\mathbf{Z}/p^n\mathbf{Z}$-extensions of $K$. Since this theory is hard to find in literature, we have included a complete treatment in the paper. In the second part of the paper, we study $\mathbf{Z}_p$-extensions of a local field $K=k((T))$ of characteristic $p>0$ where $k$ is a finite field. Local class field theory and Artin-Schreier-Witt theory give us the Schmid-Witt symbol $$[\ ,\ ): W(K)/\wp W(K) \times \widehat{K^*} \to W(\mathbf{F}_p)=\mathbf{Z}_p,$$ which contains the ramification information of $\mathbf{Z}_p$-extensions of $K$. We present a new simplified formula for $[\ ,\ )$. This formula allows one to compute ramification groups, conductors and discriminants in an easy way. In the third part, we study $\mathbf{Z}_p$-extensions of global function fields over a finite field. First, we give a formula for computing the genus in such a tower. We show that a previously obtained lower bound for the genus growth in a $\mathbf{Z}_p$-extension is incorrect and we give a sharp lower bound. We also study when the genus behaves in a `stable' way. Finally, we find unique representatives of $\mathbf{Z}_p$-extensions of the rational function field $k(X)$, and compute the genus in such a tower.

math.NT↗
arXiv2016

Deep Holes in Reed-Solomon Codes Based on Dickson Polynomials

For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of $\mathcal{C}$ is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear with a special uniformity. To find families of received words that are not deep holes, we reduce to a subset sum problem (or equivalently, a Dickson polynomial-variation of Waring's problem) and find solution conditions by applying an argument using estimates on character sums indexed over the evaluation set.

math.NT↗
arXiv2016

Newton slopes for Artin-Schreier-Witt towers

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total Galois group $\mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.

math.NT↗
arXiv2016

Power moments of Kloosterman sums

In this paper we give an essential treatment for estimating arbitrary integral power moments of Kloosterman sums over the residue class ring. For prime moduli we derive explicit estimates, and for prime-power moduli we prove concrete formulas using computations with Igusa zeta functions.

math.NT↗
arXiv2015

Counting polynomial subset sums

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that \begin{align*} \sum_{x\in S} f(x)=b. \end{align*} In this paper, we establish several asymptotic formulas for $N_f(D,k, b)$, depending on the nature of the ring $R$ and $f$. For $R=\mathbb{Z}_n$, let $p=p(n)$ be the smallest prime divisor of $n$, $|D|=n-c \geq C_dn p^{-\frac 1d }+c$ and $f(x)=a_dx^d +\cdots +a_0\in \mathbb{Z}[x]$ with $(a_d, \dots, a_1, n)=1$. Then $$\left| N_f(D, k, b)-\frac{1}{n}{n-c \choose k}\right|\leq {δ(n)(n-c)+(1-δ(n))(C_dnp^{-\frac 1d}+c)+k-1\choose k},$$ partially answering an open question raised by Stanley \cite{St}, where $δ(n)=\sum_{i\mid n, μ(i)=-1}\frac 1 i$ and $C_d=e^{1.85d}$. Furthermore, if $n$ is a prime power, then $δ(n) =1/p$ and one can take $C_d=4.41$. For $R=\mathbb{F}_q$ of characteristic $p$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $d$ not divisible by $p$ and $D\subseteq \mathbb{F}_q$ with $|D|=q-c\geq (d-1)\sqrt{q}+c$. Then $$\left| N_f(D, k, b)-\frac{1}{q}{q-c \choose k}\right|\leq {\frac{q-c}{p}+\frac {p-1}{p}((d-1)q^{\frac 12}+c)+k-1 \choose k}.$$ If $f(x)=ax+b$, then this problem is precisely the well-known subset sum problem over a finite abelian group. Let $G$ be a finite abelian group and let $D\subseteq G$ with $|D|=|G|-c\geq c$. Then $$\left| N_x(D, k, b)-\frac{1}{|G|}{|G|-c \choose k}\right|\leq {c + (|G|-2c)δ(e(G))+k-1 \choose k},$$ where $e(G)$ is the exponent of $G$ and $δ(n)=\sum_{i\mid n, μ(i)=-1}\frac 1 i$. In particular, we give a new short proof for the explicit counting formula for the case $D=G$.

math.NT↗
arXiv2015

Index bounds for character sums with polynomials over finite fields

We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings of small indices. As an application, we also give some general bounds for numbers of solutions of some Artin-Schreier equations and mininum weights of some cyclic codes.

math.NT↗
arXiv2015

On the minimum distance of elliptic curve codes

Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as AG codes. For AG codes of genus $0$ (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [3] says that the minimum distance problem is already \np-hard (under \rp-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus $1$). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least $2/3$ of the group order). Our method is purely combinatorial and based on a new sieving technique from the first two authors [8]. This method also proves a significantly stronger version of the MDS (maximum distance separable) conjecture for ECAG codes.

cs.IT↗
arXiv2014

Linearized Wenger graphs

Motivated by recent extensive studies on Wenger graphs, we introduce a new infinite class of bipartite graphs of the similar type, called linearized Wenger graphs. The spectrum, diameter and girth of these linearized Wenger graphs are determined.

math.CO↗
arXiv2013

Traps to the BGJT-Algorithm for Discrete Logarithms

In the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thom{é}, a quasi-polynomial time algorithm (QPA) is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular, when the field is not a Kummer extension. We believe that the basic idea behind the new approach should still work, and propose a fix to the algorithm in non-Kummer cases, without altering the quasi-polynomial time complexity. The modified algorithm is also heuristic. Further study is required in order to fully understand the effectiveness of the new approach.

math.NT↗
arXiv2013

Stopping Sets of Algebraic Geometry Codes

Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let $C$ be an $[n,k]$ linear code over $\f$ with parity-check matrix $H$, where the rows of $H$ may be dependent. Let $[n]=\{1,2,...,n\}$ denote the set of column indices of $H$. A \emph{stopping set} $S$ of $C$ with parity-check matrix $H$ is a subset of $[n]$ such that the restriction of $H$ to $S$ does not contain a row of weight 1. The \emph{stopping set distribution} $\{T_{i}(H)\}_{i=0}^{n}$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Denote $H^{*}$ the parity-check matrix consisting of all the non-zero codewords in the dual code $C^{\bot}$. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix $H^*$. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed-Solomon codes, it is easy to determine all the stopping sets. Then we consider AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then the stopping sets, the stopping set distribution and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of AG codes from elliptic curves.

cs.IT↗
arXiv2013

Algebraic Cayley Graphs over Finite Fields

A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of Chung.

math.CO↗
arXiv2012

Value sets of polynomial maps over finite fields

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

math.NT↗
arXiv2012

L-functions of p-adic characters

We define a p-adic character to be a continuous homomorphism from 1 + t\Fq[[t]] to \Zp^*. We use the ring of big Witt vectors over Fq to exhibit a bijection between p-adic characters and sequences (c_i) of elements in Zq, indexed by natural numbers relatively prime to p, and which converge to zero p-adically. To such a p-adic character we associate an L-function, and we prove that this L-function is p-adic meromorphic if the corresponding sequence (c_i) is overconvergent. If more generally the sequence is c\log-convergent, we show that the associated L-function is meromorphic in the open disk of radius q^c. Finally, we exhibit examples of c\log-convergent sequences with associated L-functions which are not meromorphic in any disk of radius greater than q^c.

math.NT↗
arXiv2011

Counting Value Sets: Algorithm and Complexity

Let $p$ be a prime. Given a polynomial in $\F_{p^m}[x]$ of degree $d$ over the finite field $\F_{p^m}$, one can view it as a map from $\F_{p^m}$ to $\F_{p^m}$, and examine the image of this map, also known as the value set. In this paper, we present the first non-trivial algorithm and the first complexity result on computing the cardinality of this value set. We show an elementary connection between this cardinality and the number of points on a family of varieties in affine space. We then apply Lauder and Wan's $p$-adic point-counting algorithm to count these points, resulting in a non-trivial algorithm for calculating the cardinality of the value set. The running time of our algorithm is $(pmd)^{O(d)}$. In particular, this is a polynomial time algorithm for fixed $d$ if $p$ is reasonably small. We also show that the problem is #P-hard when the polynomial is given in a sparse representation, $p=2$, and $m$ is allowed to vary, or when the polynomial is given as a straight-line program, $m=1$ and $p$ is allowed to vary. Additionally, we prove that it is NP-hard to decide whether a polynomial represented by a straight-line program has a root in a prime-order finite field, thus resolving an open problem proposed by Kaltofen and Koiran in \cite{Kaltofen03,KaltofenKo05}.

math.NT↗