Irreducibility Testing Research Articles

A quasi-linear irreducibility test in $$\mathbb{K}[[x]][y]$$

We provide an irreducibility test in the ring $$\mathbb{K}[[x]][y]$$ whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial $$F$$ is square-free and $$\mathbb{K}$$ is a perfect field of characteristic not dividing $$\deg(F)$$ . The algorithm uses the theory of approximate roots and may be seen as a generalisation of Abhyankar's irreducibility criterion to the case of non-algebraically closed residue fields.

An Irreducibility Test for Polynomials whose Coefficients are Algebraic Integers

For a non-zero algebraic integer α, let ℚ(α) denote the simple extension of the field of rational numbers ℚ. ℤ[α] is the smallest subring of ℚ(α) containing both ℤ and α. In this article, we present an account for testing irreducibility of a given polynomial with coefficients in ℤ[α] over the field ℚ(α).

ALGORITHMS FOR FACTORIZATION AND IRREDUCIBILITY TESTING OF POLYNOMIALS USING ELLIPTIC CURVES

In this article irreducibility criteria and factorization algorithms are formulated and proved, that are generalizations of correspondent criteria and algorithms for integers. Generalization of Lenstra’s theorem is of especial interest. It provides correct calculations of multiple points on elliptic curve that is used in Lenstra algorithm. Note that the generalization of this theorem in case of finite field of characteristic 2 is completely different from the classical case and is very non-trivial. Using the criteria obtained, we construct a series of algorithms for factoring and testing of irreducibility of polynomials over finite field. We should note that their performance isn’t their main advantage, but the main advantage is using some abelian group built on elliptic curve. So in case of failure of algorithm we can just chose the other elliptic curve over the corresponding field and repeat the algorithm. These algorithms have no large advantage in speed, like their preimages – analogical algorithms for integers, but they demonstrate new approaches to the solutions of important problem of irreducibility testing and also are very interesting from mathematical point of view.

Factoring formal power series over principal ideal domains

We provide an irreducibility test and factoring algorithm (with some qualificiations) for formal power series in the unique factorization domain R((X)), where R is any principal ideal domain. We also classify all integral domains arising as quotient rings of R((X)). Our main tool is a generalization of the p-adic Weierstrass preparation theorem to the context of complete filtered commutative rings.

Modular Las Vegas algorithms for polynomial absolute factorization

Let f(X,Y)∈Z[X,Y] be an irreducible polynomial over Q. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f, or more precisely, of f modulo some prime integer p. The same idea of choosing a p satisfying some prescribed properties together with LLL is used to provide a new strategy for absolute factorization of f(X,Y). We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to construct the algebraic extension containing one absolute factor of a polynomial of degree up to 400.

Irreducibility testing of finite nilpotent linear groups

We describe an algorithm for irreducibility testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For a reducible group, our algorithm constructs a proper submodule. An implementation in Magma is publicly available.

An empirical study of cache-oblivious polygon indecomposability testing

We implement and undertake an empirical study of the cache-oblivious variant of the polygon indecomposability testing algorithm of Gao and Lauder, based on a depth-first search (DFS) traversal of the computation tree. According to Abu Salem, the cache-oblivious variant exhibits improved spatial and temporal locality over the original one, and its spatial locality is optimal. Our implementation revolves around eight different variants of the DFS-based algorithm, tailored to assess the trade-offs between computation and memory performance as originally proposed by Abu Salem. We analyse performance sensitively to manipulations of the several parameters comprising the input size. We describe how to construct suitably random families of input that solicit such variations, and how to handle redundancies in vector computations at no asymptotic increase in the work and cache complexities. We report extensively on our experimental results. In all eight variants, the DFS-based variant achieves excellent performance in terms of L1 and L2 cache misses as well as total run time, when compared to the original variant of Gao and Lauder. We also benchmark the DFS variant against the powerful computer algebra system MAGMA, in the context of bivariate polynomial irreducibility testing using polygons. For sufficiently high degree polynomials, MAGMA either runs out of memory or fails to terminate after about 4 h of execution. In contrast, the DFS-based version processes such input using a couple of seconds. Particularly, we report on absolute irreducibility testing of bivariate polynomials of total degree reaching 19,000 in about 2 s for the DFS variant, using a single processor.

Towards an efficient Meat-axe algorithm using f-cyclic matrices: The density of uncyclic matrices in [formula omitted

An element X in the algebra M ( n , F ) of all n × n matrices over a field F is said to be f-cyclic if the underlying vector space considered as an F [ X ] -module has at least one cyclic primary component. These are the matrices considered to be “good” in the Holt–Rees version of Norton's irreducibility test in the Meat-axe algorithm. We prove that, for any finite field F q , the proportion of matrices in M ( n , F q ) that are “not good” decays exponentially to zero as the dimension n approaches infinity. Turning this around, we prove that the density of “good” matrices in M ( n , F q ) for the Meat-axe depends on the degree, showing that it is at least 1 − 2 q ( 1 q + 1 q 2 + 2 q 3 ) n for q ⩾ 4 . We conjecture that the density is at least 1 − 1 q ( 1 q + 1 2 q 2 ) n for all q and n, and confirm this conjecture for dimensions n ⩽ 37 . Finally we give a one-sided Monte Carlo algorithm called Is f Cyclic to test whether a matrix is “good,” at a cost of O ( Mat ( n ) log n ) field operations, where Mat ( n ) is an upper bound for the number of field operations required to multiply two matrices in M ( n , F q ) .

Parallel methods for absolute irreducibility testing

A heuristic algorithm for testing absolute irreducibility of multivariate polynomials over arbitrary fields using Newton polytopes was proposed in Gao and Lauder (Discrete Comput. Geom. 26:89–104, [2001]). A preliminary implementation by Gao and Lauder (2003) established a wide range of families of low degree and sparse polynomials for which the algorithm works efficiently and with a high success rate. In this paper, we develop a BSP variant of the absolute irreducibility testing via polytopes, with the aim of producing a more memory and run-time efficient method that can provide wider ranges of applicability, specifically in terms of the degrees of the input polynomials. In the bivariate case, we describe a balanced load scheme and a corresponding data distribution leading to a parallel algorithm whose efficiency can be established under reasonably realistic conditions. This is later incorporated in a doubly parallel algorithm in the multivariate case that achieves similar scalable performance. Both parallel models are analyzed for efficiency, and the theoretical analysis is compared to the performance of our experiments. In the empirical results we report, we achieve absolute irreducibility testing for bivariate and trivariate polynomials of degrees up to 30,000, and for low degree multivariate polynomials with more than 3,000 variables. To the best of our knowledge, this sets a world record in establishing absolute irreducibility of multivariate polynomials.

Pseudoirreducible Polynomials: Probabilistic Irreducibility Testing

Polynomial analogs of pseudoprime numbers are created (Fermat pseudoprimes, Euler pseudoprimes, and strong pseudoprimes). Some of their properties and interrelations are described. Efficient probabilistic algorithms of irreducibility testing are given that are analogs of the Fermat, Solovay-Strassen, and Miller-Rabin algorithms.

Analysis of Rabin's irreducibility test for polynomials over finite fields

AbstractWe give a precise average‐case analysis of Rabin's algorithm for testing the irreducibility of polynomials over finite fields. The main technical contribution of the article is the study of the probability that a random polynomial of degree n contains an irreducible factor of degree dividing several maximal divisors of the degree n. We then study the expected value and the variance of the number of operations performed by the algorithm. We present an exact analysis when n=p1 and n=p1p2 for p1, p2 prime numbers, and an asymptotic analysis for the general case. Our method generalizes to other algorithms that deal with similar divisor conditions. In particular, we analyze the average‐case number of operations for two variants of Rabin's algorithm, and determine the ordering of prime divisors of n that minimizes the leading factor. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 525–551, 2001

Decomposition of Polytopes and Polynomials

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.

Irreducibility testing over local fields

The purpose of this paper is to describe a method to determine whether a bivariate polynomial with rational coefficients is irreducible when regarded as an element in Q((x))[y], the ring of polynomials with coefficients from the field of Laurent series in x with rational coefficients. This is achieved by computing certain associated Puiseux expansions, and as a result, a polynomial-time complexity bound for the number of bit operations required to perform this irreducibility test is computed.

Analysis of Ben-Or's polynomial irreducibility test

We give a precise average-case analysis of Ben-Or's algorithm for testing the irreducibility of polynomials over finite fields. First, we study the probability that a random polynomial of degree n over q contains no irreducible factors of degree less than m, 1≤m≤n. The results are given in terms of the Buchstab function. Then, we compute the expectation and variance of the smallest degree among the irreducible factors of a random polynomial. The analysis of Ben-Or's algorithm readily follows from this expectation. We also compute the probability of a polynomial being irreducible when it has no irreducible factors of degree less than m, 1≤m≤n. This probability is useful in the analysis of some algorithms for factoring polynomials over finite fields. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 439–456, 1998

Analysis of Ben‐Or's polynomial irreducibility test

Analysis of Ben‐Or's polynomial irreducibility test

Effective Noether Irreducibility Forms and Applications

Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class NC. The factorization results also extend to the case where the coefficient field is a function field.

A heuristic irreducibility test for univariate polynomials

This paper describes a heuristic irreducibility test for univariate polynomials over the integers. The irreducibility test is based on finding a prime evaluation of a polynomial which, under appropriate conditions, is a witness to the irreducibility of the polynomial. The irreducibility test has been implemented as part of the polynomial factorization package available in the Maple symbolic algebra system. Several implementation details are important for the practicality of this approach, in particular, a probabilistic primality test is used. Therefore, the irreducibility test does not constitute a proof of irreducibility. Timing results are presented for a range of sample problems. The empirical results obtained compare well with the Berlekamp-Hensel algorithm for factoring polynomials, as it is implemented in the MAPLE, MACSYMA and REDUCE symbolic algebra systems.

Approximate factorization of multivariate polynomials and absolute irreducibility testing

A concept of approximate factorization of multivariate polynomial is introduced and an algorithm for approximate factorization is presented. The algorithm handles polynomials with complex coefficients represented approximately, hence it can be used to test the absolute irreducibility of multivariate polynomials. The algorithm works as follows: given a monic square-free polynomialF(x,y,…,z), it calculates the roots ofF(x,yo, ...,z0) numerically, whereyo, … z0 are suitably chosen numbers, then it constructs power series F1, …, Fn such thatF(x,y, …, z) ∈F1 (x,y., ...,z)...Fn(x,y, ...z) (mod Se+2), where n=degx (F),S=(y-y0, ...,z-z0), ande=max{degy(F), …, degx (F)}; finally it finds the approximate divisors ofF as products of elements of {F1, …,Fn}.

Deterministic irreducibility testing of polynomials over large finite fields

We present a sequential deterministic polynomial-time algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility testing combined with Berlekamp's algorithm.

Effective Hilbert irreducibility

In this paper we prove by entirely elementary means a very effective version of the Hilbert irreducibility theorem. We then apply our theorem to construct a probabilistic irreducibility test for sparse multivariate polynomials over arbitrary perfect fields. For the usual coefficient fields the test runs in polynomial time in the input size.