Related papers: Newton polygons and p-integral bases
The aim of this study is to provide a perspective to help understand the singular average operator over polynomial hypersurfaces. In particular, this perspective will provide brevity and the possibility of generalizing previous results…
We develop an algorithm for recursively constructing orthogonal multiplet bases for the color space of QCD, for any order of partons and any Nc. This recipe is then applied for explicitly constructing some of these bases. Using the bases, a…
In this paper, by using matix representation for quaternions andoctonions, we provide a procedure to obtain some example of k potent matrices of order 4 or 8, over the real field or over the field ZP, with p a prime number.
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces…
In addition to rather complicated general methods it is interesting and valuable to develop fast efficient methods for calculating generators of power integral bases in special types of number fields. We consider sextic fields containing a…
Let $F$ be an algebraically closed field of characteristic $p>0$. In this paper we develop methods to represent arbitrary elements of $F[t]$ as sums of perfect $k$-th powers for any $k\in\mathbb{N}$ relatively prime to $p$. Using these…
We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution…
This paper first introduces the concept of p-adic number and field. Then it develops the p-adic integration and applied it to solve p-adic Schrodinger equations.
Using elementary methods, we prove new formulas for $\operatorname{pp}(n)$, the number of plane partitions of $n$, $\operatorname{pp}_r(n)$, the number of plane partitions of $n$ with at most $r$ rows, $\operatorname{pp}^s(n)$, the number…
The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…
J. Harrington and L. Jones characterized monogenity of four new parametric families of quartic polynomials with various Galois groups. A short time later P. Voutier added a cyclic family. In this note we intend to describe all generators of…
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…
This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
Following Faber-Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $ \mathbb{P}^1 $ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are…
By using techniques of poset representation theory, we present a formula for the number of partitions of a positive integer into three polygonal numbers.
We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our…
A construction of polytopes is given based on integers. These geometries are constructed through a mapping to pure numbers and have multiple applications, including statistical mechanics and computer science. The number form is useful in…
We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient…