Related papers: A Division Algorithm Approach to $p$-Adic Sylveste…
In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…
Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision.…
In this paper, we develop an algorithm for computing Coleman--Gross (and hence Nekov\'a\v{r}) $p$-adic heights on hyperelliptic curves over number fields with arbitrary reduction type above $p$. This height is defined as a sum of local…
We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective…
We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus…
Reasoning with defeasible and conflicting knowledge in an argumentative form is a key research field in computational argumentation. Reasoning under various forms of uncertainty is both a key feature and a challenging barrier for automated…
For solving the continuous Sylvester equation, a class of the multiplicative splitting iteration method is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations…
This article presents a new method for calculating square roots in GF(p) by exponentiating in GF(p^3) or equivalently modulo irreducible cubic polynomials. This algorithm is in some ways similar to the Cipolla-Lehmer algorithm which is…
We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…
Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's {\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whose degree is $2p$, as a polynomial function of irrationals of degrees $\le p$. In certain…
We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of…
In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given…
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 develop a continued fraction algorithm in finite extensions of $\Q_p$ generalising certain algorithms in $\Q_p$, and prove the finiteness property for certain small degree extensions. We also discuss the metrical properties of the…
We calculate extensions between certain irreducible admissible representations of p-adic groups.
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…
In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.
We use the theory of motivic integration in order to give a geometric explanation of the behavior of some p-adic integrals.