相关论文: Factoring bivariate sparse (lacunary) polynomials
We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is…
Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an…
In this paper, we provide the degree distribution of irreducible factors of the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial. We…
We present a new deterministic algorithm for the sparse Fourier transform problem, in which we seek to identify k << N significant Fourier coefficients from a signal of bandwidth N. Previous deterministic algorithms exhibit quadratic…
In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
Robust mean estimation is one of the most important problems in statistics: given a set of samples in $\mathbb{R}^d$ where an $\alpha$ fraction are drawn from some distribution $D$ and the rest are adversarially corrupted, we aim to…
Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…
Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following:…
Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…
New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…
Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $\nu_p(q-1)\ge \nu_{p}(e)+\nu_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine…
Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…
We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…
We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$…
We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of…
We show that Reed-Solomon codes of dimension $k$ and block length $n$ over any finite field $\mathbb{F}$ can be deterministically list decoded from agreement $\sqrt{(k-1)n}$ in time $\text{poly}(n, \log |\mathbb{F}|)$. Prior to this work,…
Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaqui\'e and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the…
We introduce a novel algorithm that computes the $k$-sparse principal component of a positive semidefinite matrix $A$. Our algorithm is combinatorial and operates by examining a discrete set of special vectors lying in a low-dimensional…
In this paper we describe a quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials. The algorithm is nondeterministic and its expected resource requirements are polylogarithmic in…