Related papers: Performance of Buchberger's Improved Algorithm usi…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
We call an integer N>1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method…
We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gr\"obner bases. We present a novel scalable algorithm which combines the two approaches and leads to the…
In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
Experiences with the implementation of strong Gr\"obner bases respectively standard bases for polynomial rings over principal ideal rings are explained: different strategies for creating the pair set, methods to avoid coefficient growth and…
In this summary we discuss two new algorithms for Grover's unsorted database search problem that claimed to have reached exponential speedup over Grover's original algorithm. One is in the quantum setting with "power queries" that allow for…
The reduction of Feynman integrals to master integrals is an algebraic problem that requires algorithmic approaches at the modern level of calculations. Straightforward applications of the classical Buchberger algorithm to construct…
We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order…
The aim of this article is to introduce standard bases of ideals in polynomial rings with respect to a class of orderings which are not necessarily semigroup orderings. Our approach generalises the concept of standard bases with respect to…
The complexity of Gr\"{o}bner computations has inspired many improvements to Buchberger's algorithm over the years. Looking for further insights into the algorithm's performance, we offer a threaded implementation of classical Buchberger's…
Recently, a novel method based on coding partitions [1]-[4] has been used to derive power series expansions to previously intractable problems. In this method the coefficients at $k$ are determined by summing the contributions made by each…
Prime numbers are fundamental in number theory and play a significant role in various areas, from pure mathematics to practical applications, including cryptography. In this contribution, we introduce a multithreaded implementation of the…
Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their…
It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…