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An algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix. The algorithm is especially suited for the evaluation of canonical traces in determinant…
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…
We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our approach consists of a combination of the $p$-adic approximation…
We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a…
We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional…
We produce a collection of families of curves, whose point count statistics over F_p becomes Gaussian for p fixed. In particular, the average number of F_p points on curves in these families tends to infinity.
We use recent results on algorithms for Markov decision problems to show that a canonical form for a generalized P-matrix can be computed, in some important cases, by a strongly polynomial algorithm.
We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known…
In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row…
Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer…
Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator $\mathcal{A}$ given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially,…
We address complexity issues for linear differential equations in characteristic $p>0$: resolution and computation of the $p$-curvature. For these tasks, our main focus is on algorithms whose complexity behaves well with respect to $p$. We…
A polynomial-time algorithm for computing the permanent in any field of characteristic 3 is presented in this article. The principal objects utilized for that purpose are the Cauchy and Vandermonde matrices, the discriminant function and…
An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.
This article presents a numerical illustration of a recently proposed strongly polynomial-time algorithm for the general linear programming (LP) problem. Each iteration of the proposed algorithm consists of two Gauss-Jordan pivoting…
In algebraic geometry, enumerating or finding superspecial curves in positive characteristic $p$ is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic…
Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach…