Improved Polynomial Remainder Sequences for Ore Polynomials

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients

Skew-cyclic codes

We generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew-cyclic codes, this new class of codes allows to systematically search for codes with good properties. We give many examples of codes which improve the previously best known linear codes

A general framework for Noetherian well ordered polynomial reductions

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper.Comment: 36 pages. New title and substantial improvements to the presentation according to the comments of the reviewer

Survey on counting special types of polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f is decomposable if f = g o h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f. We present a classification of all collisions at degree n = p^2 which yields an exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann (editors), Computer Algebra and Polynomials, Lecture Notes in Computer Scienc

Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof

In this article, we provide a comprehensive historical survey of 183 different proofs of famous Euclid's theorem on the infinitude of prime numbers. The author is trying to collect almost all the known proofs on infinitude of primes, including some proofs that can be easily obtained as consequences of some known problems or divisibility properties. Furthermore, here are listed numerous elementary proofs of the infinitude of primes in different arithmetic progressions. All the references concerning the proofs of Euclid's theorem that use similar methods and ideas are exposed subsequently. Namely, presented proofs are divided into 8 subsections of Section 2 in dependence of the methods that are used in them. {\bf Related new 14 proofs (2012-2017) are given in the last subsection of Section 2.} In the next section, we survey mainly elementary proofs of the infinitude of primes in different arithmetic progressions. Presented proofs are special cases of Dirichlet's theorem. In Section 4, we give a new simple "Euclidean's proof" of the infinitude of primes.Comment: 70 pages. In this extended third version of the article, 14 new proofs of the infnitude of primes are added (2012-2017

Zeros with multiplicity, Hasse derivatives and linear factors of general skew polynomials

In this work, multiplicities of zeros of general skew polynomials are studied. Two distinct definitions are considered: First, $a$ is said to be a zero of $F$ of multiplicity $r$ if $(x-a)^r$ divides $F$ on the right; second, $a$ is said to be a zero of $F$ of multiplicity $r$ if some skew polynomial $P = (x-a_r) \cdots (x-a_2) (x-a_1)$, having $a_1 = a$ as its only right zero, divides $F$ on the right. The first notion was considered earlier, while the second one was recently introduced by Bolotnikov over the quaternions. Neither of these two notions implies the other for general skew polynomials. We show that, in the first case, Lam and Leroy's concept of P-independence does not behave naturally, whereas a union theorem, as stated by Lam, still holds. In contrast, we show that P-independence for the second notion of multiplicities behaves naturally. As a consequence, we provide extensions of classical commutative results to general skew polynomials. These include the upper bound on the number of (P-independent) zeros (counting multiplicities) of a skew polynomial by its degree, and the equivalence of P-independence, the solvability of Hermite interpolation and the invertibility of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse derivatives). We provide characterizations of skew polynomials of the form $P = (x-a_r) \cdots (x-a_2) (x-a_1)$ having a single right zero, assuming conjugacy classes are algebraic. Based on these, we explicitly describe such skew polynomials in particular cases of interest, recovering Bolotnikov's results over the quaternions as a particular case

Resultant-based Elimination in Ore Algebra

We consider resultant-based methods for elimination of indeterminates of Ore polynomial systems in Ore algebra. We start with defining the concept of resultant for bivariate Ore polynomials then compute it by the Dieudonne determinant of the polynomial coefficients. Additionally, we apply noncommutative versions of evaluation and interpolation techniques to the computation process to improve the efficiency of the method. The implementation of the algorithms will be performed in Maple to evaluate the performance of the approaches.Comment: An updated (and shorter) version published in the SYNASC '21 proceedings (IEEE CS) with the title "Resultant-based Elimination for Skew Polynomials