Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\widetilde{O}(n^{3/2}\log q + n \log^2 q)$ time to factor polynomials of degree $n$ over the finite field $\mathbb{F}_q$ with $q$ elements. A significant open problem is if the $3/2$ exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than $3/2$ would yield an algorithm for polynomial factorization with exponent better than $3/2$

On solving systems of random linear disequations

An important subcase of the hidden subgroup problem is equivalent to the shift problem over abelian groups. An efficient solution to the latter problem would serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the shift problem is reduction to solving systems of certain random disequations in finite abelian groups. The random disequations are actually generalizations of linear functions distributed nearly uniformly over those not containing a specific group element in the kernel. In this paper we give an algorithm which finds the solutions of a system of N random linear disequations in an abelian p-group A in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.Comment: 13 page

Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits

We make an observation which doubles the exponent of distribution in certain Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and Kontorovich-Oh. As a consequence, we decrease the known bounds on the saturation numbers in these problems.Comment: 24 page

On the exponent of a primitive, minimally strong digraph

AbstractWe consider n × n primitive nearly reducible matrices for n⩾5. As defined by Ross, let e(n) be the least integer such that no such n × n matrix has this integer as its exponent. The investigation of e(n) is the first open problem in Ross's paper. Here we offer a method to compute e(n) for small n. Then we generalize Ross's estimate, which is e(n) > n+1, to e(n) > (p+1)(n−p) for n⩾2p+1, and p⩾11 a prime less than 100,000. There are extant various estimates and conjectures concerning the difference of successive primes. If one of the most hopeful of these conjectures be true, then our lower bound for e(n) holds for all p⩾11 and in fact we obtain e(n) > n24−(n2)32

The exponent set of symmetric primitive (0, 1) matrices with zero trace

AbstractWe prove that the exponent set of symmetric primitive (0, 1) matrices with zero trace (the adjacency matrices of the simple graphs) is {2,3,…,2n−4}⧹S, where S is the set of all odd numbers in {n−2,n−1,…,2n−5}. We also obtain a characterization of the symmetric primitive matrices with zero trace whose exponents attain the upper bound 2n−4