Related papers: Fast Coefficient Computation for Algebraic Power S…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same…
We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier…
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…
In this paper, we extend our previous work on the power series method for computing backstepping kernels. Our first contribution is the development of initial steps towards a MATLAB toolbox dedicated to backstepping kernel computation. This…
We construct fast algorithms for evaluating transforms associated with families of functions which satisfy recurrence relations. These include algorithms both for computing the coefficients in linear combinations of the functions, given the…
Christol's theorem characterises algebraic power series over finite fields in terms of finite automata. In a recent article, Bridy develops a new proof of Christol's theorem by Speyer, to obtain a tight quantitative version, that is, to…
Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to…
We present new results and an algorithm for standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring in finitely many variables over a field K. The algorithm provides a…
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for…
In connection with our previous work on semi-galois categories, this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series over finite field is algebraic…
This work formalizes efficient Fast Fourier-based multiplication algorithms for polynomials in quotient rings such as $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$, with $n$ a power of 2 and $m$ a non necessarily prime integer. We also present a…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
A new type of algorithms is presented that combine the advantages of quantum and classical ones. Those combined advantages along with aspects of Geometric Algebra that open possibilities unavailable to both of these computations are…
Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This…
In this paper we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristics $p$. Let $g$ be such a series, then $g$ has a fixed point at the origin and the corresponding…
We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the…
We present an algorithm for factoring linear differential operators with coefficients in a finite separable extension of F p (x). Our methods rely on specific tools arising in positive characteristic: p-curvature, structure of simple…
We examine the behavior of the coefficients of powers of polynomials over a finite field of prime order. Extending the work of Allouche-Berthe, 1997, we study a(n), the number of occurring strings of length n among coefficients of any power…
In this article we give the details of an effective point counting algorithm for genus two curves over finite fields of characteristic three. The algorithm has an application in the context of curve based cryptography. One distinguished…
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a…