Reconstructing rational functions with FireFly

Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (outliers). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers (cleans up data) through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.

In this paper, we consider rational functions [Formula: see text] with some minor restrictions over the finite field [Formula: see text] where [Formula: see text] for some prime [Formula: see text] and positive integer [Formula: see text]. We establish a sufficient condition for the existence of a pair [Formula: see text] of primitive normal elements in [Formula: see text] over [Formula: see text] Moreover, for [Formula: see text] and rational functions [Formula: see text] with quadratic numerators and denominators, we explicitly find that there are at most [Formula: see text] finite fields [Formula: see text] in which such a pair [Formula: see text] of primitive normal elements may not exist.

We describe a coarse-grain parallel approach for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. We have performed an outer loop parallelization that works well in conjunction with a black box abstraction for the coefficient matrix. Our implementation can be run on a network cluster of UNIX workstations as well as on an SP-2 multiprocessor. Task distribution and management are effected through MPI and other packages. Fault tolerance, checkpointing, and recovery are incorporated. Detailed timings are presented for experiments with systems that arise in RSA challenge integer factoring efforts. For example, we can solve a 252,222 × 252,222 system with about 11.04 million nonzero entries over the Galois field with two elements using four processors of an SP-2 multiprocessor, in about 26.5 hours CPU time.

A two-dimensional (2D) signal or image can be represented by a rational complex function. Also, a two-dimensional, linear and shift invariant system can be modeled by a two-dimensional auto-regressive, moving-averaging (ARMA) process. A recently proposed linear technique for image or 2D system representation, by using a rational function, is briefly presented. This method, which is based on the recursive like structure or equationary model, can approximate the unknown 2D system transfer function, or the transform of an image, in the frequency domain in the least-square sense. The authors discuss the important existence, uniqueness and convergence properties associated with this technique. In addition, design examples are also given to show the excellent performance of the algorithm. >

Let \( \mathbb{F}_q \) be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over \( \mathbb{F}_q \) and \( \mathbb{F}_q \) be the rational function field over \( \mathbb{F}_q \). We seek to understand the structure of the rational invariant subfield \( \mathbb{F}_q \). In this paper, we prove that \( \mathbb{F}_q \) is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases.

Rational functions over finite fields having continued fraction expansions with linear partial quotients

We present a well-parallelizable algorithm which, taking a straight-line program for the evaluation of a vectorial field of rational functions of Q(X 1,...,X n ) as input, decides whether they allow a rational potential function and, in case of affirmative answer, computes it as output. We introduce a mixed model of representation of polynomials to allow the application of integration techniques and show how to perform some basic operations with it. The algorithm is presented as a family of arithmetic networks of polynomial size and poly logarithmic depth in the degree of the occurring polynomials.KeywordsRational FunctionPartial FractionGreat Common DivisorProbabilistic AlgorithmParallel TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Statement of results We let F be the finite field with two elements, and let K be the field F((x')) of all formal power series in x-' over F. Every element of K not in the field of rational functions F(x) has a unique infinite continued fraction expansion f = a. + 1/(al + 1/(a2 + * * *)), or more briefly f = [a0; a,, a2, * *.* ],

We present algorithms to construct an efficient black box for the pseudoinverse, At, of a black box matrix A. This in also known as the Moore-Penrose inverse of A. For the system Ax = b over a subfield of the complex numbers: the vector x = Atb is the least squares solution having the least norm. When we say that A is a black box matrix we mean simply that methods are given to compute the matrix-vector products Au and uTA, for vectors u and vectors v of compatible length. No other assumptions are made about the structure or representation of the matrix.The entries may be integers or rational numbers or elements of a finite field. For the integer or rational case the method uses homomorphic images, and is based on an algorithm capable of computing the black box for the pseudoinverse, when it exists, over an arbitrary field. We discuss the asymptotic costs and some variants that are likely to be useful in practice.Over a finite field the solution, x = Atb, is uniquely defined when it exists. The ability to produce this particular solution is a useful addition to the range of ways solution to singular linear systems may be offered in the black box model.

Let K = ¥q(T the rational function field over the finite field ¥q (T: indeterminate), and A=¥q[T]. For a non-zero ideal n of A, we can define a smooth proper geometrically connected curve X0(n) over K, called the Drinfeld modular curve of Hecke type with conductor n. In this article, we define the Eisenstein quotient J of the Jacobian variety / of Jf0(n) for n maximal and investigate its arithmetic properties. One of the main results is as follows :

Uncertainty principle and sparse reconstruction in pairs of orthonormal rational function bases

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Tests for Permutation Functions