Irreducible Factors Research Articles

Multivariate to bivariate reduction for noncommutative polynomial factorization

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,1.Given an n-variate noncommutative polynomial f∈F〈X〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g∈F〈x,y〉; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.The reduction works both in the white-box and the black-box setting.2.We show over the field of rationals that bivariate linear matrix factorization problem for 4×4 matrices is at least as hard as factoring square-free integers and for 3×3 matrices it is in polynomial time.

Some factorization results for bivariate polynomials

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials f(x, y) over an arbitrary field K . Our results rely on information on the degrees of the coefficients of f, and on information on the factorization of the constant term and of the leading coefficient of f, viewed as a polynomial in y with coefficients in K [ x ] . In particular, we provide a generalization of the bivariate version of Perron’s irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of f in an algebraic closure of K ( x ) .

Long-term treatment with bisphosphonates in clinical practice: advantages, main problems and risks

Bisphosphonates are the main class of drugs for treatment osteoporosis (OP) and other diseases with increased bone resorption, as bisphosphonates are very effective in reducing risk of fracture. The problem of maintaining the effectiveness and possible loss of effect of bisphosphonates, as well as their safety during long-term use, remains actual Long-them therapy with bisphosphonates and it’s effects has been discussed over the past 20 years, as the risk of osteoporotic fracture may stay hight in patients with presence of irreducible risk factors (continous use of glucocorticoids etc.) despite ongoing antiosteoporotic therapy. Real clinical practice demonstrates very low patient adherence to treatment with bisphosphonates. However, observational studies have showed that treatment with bisphosphonates for more than 10 years without initiating a drug holiday can be effective for patients at high risk of fracture. Moreover, the longer therapy with bisphosphonates is continued and the later the“drug holiday”is initiated, the lower the risks of fractures of the proximal femur and clinical vertebral fractures. However, the duration of continuous bisphosphonate therapy for each patient remains at the decision of the physician and is determined individually in each case, based on the risk-benefit ratio, taking into account the patient’s risk factors for fractures and comorbid diseases.

Integral values of generating functions of recursive sequences

Suppose that a0,a1,… is an integer sequence which satisfies a recurrence relation with constant coefficients, and let T(x)=f(x)/g(x) be its generating function, where f(x) and g(x) have no common factors in Z[x]. In this article, we study the problem of finding the rational values of x such that T(x) is an integer. We say that such a number is good for the sequence. Our first main result is that if g(x) has at least two different irreducible factors, or if g(x) has a single irreducible factor of degree at least 3, then the sequence has only finitely many good values. We also study sequences of the form 0,1,… for which the recurrence relation has order 2. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them.

On the number of the irreducible factors of $ x^{n}-1 $ over finite fields

<p>Let $ \mathbb{F}_q $ be the finite field of $ q $ elements, and $ \mathbb{F}_{q^{n}} $ its extension of degree $ n $. A normal basis of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_q $ is a basis of the form $ \{\alpha, \alpha^{q}, \cdots, \alpha^{q^{n-1}}\} $. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial $ x^{n}-1 $ in $ \mathbb{F}_q $, while the latter is closely related to the cyclotomic polynomials. Denote by $ \mathfrak{F}(x^{n}-1) $ the set of all distinct monic irreducible factors of $ x^{n}-1 $ in $ \mathbb{F}_q $. The criteria for</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ |\mathfrak{F}(x^{n}-1)|\leq 2 $\end{document} </tex-math></disp-formula></p><p>have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ |\mathfrak{F}(x^{n}-1)| = s, $\end{document} </tex-math></disp-formula></p><p>where $ s $ is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for</p><p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ |\mathfrak{F}(x^{n}-1)| = 3, 4, 5. $\end{document} </tex-math></disp-formula></p>

Bi-Unitary Superperfect Polynomials over 𝔽2 with at Most Two Irreducible Factors

A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if gcd(B,A/B)=1 (resp. gcdu(B,A/B)=1), where gcdu(B,A/B) denotes the greatest common unitary divisor of B and A/B. We denote by σ**(A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F2 if the sum of all bi-unitary monic divisors of σ**(A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F2. We prove the nonexistence of odd bi-unitary superperfect polynomials over F2.

Factoring primes and sums of two squares

Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .

Factorization of composition of reciprocal polynomials with monomials

A polynomial f(x) is referred to as reciprocal if it satisfies f(x)=±xdeg⁡ff(1/x). It is established that if f(x)∈Q[x] is reciprocal and irreducible over Q, and r>0 is an odd integer, then f(xr) has an irreducible reciprocal factor over Q. Furthermore, if f(x) has a root on the unit circle, and r>0 is an arbitrary integer, then every irreducible factor of f(xr) is reciprocal. This generalizes a related result of Filaseta and Meade for reciprocal 0,1-polynomials. As an outcome of our methods, we obtain some information on the irreducible factors over Q of f(xr) for an arbitrary f(x)∈Q[x].

Stability of certain higher degree polynomials

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. A polynomial [Formula: see text] is stable over [Formula: see text] if irreducibility of [Formula: see text] implies that all its iterates are also irreducible over [Formula: see text], that is, [Formula: see text] is irreducible over [Formula: see text] for all [Formula: see text], where [Formula: see text] denotes the [Formula: see text]-fold composition of [Formula: see text]. In this paper, we study the stability of [Formula: see text] for [Formula: see text], [Formula: see text]. We show that for infinite families of [Formula: see text], whenever [Formula: see text] is irreducible, all its iterates are irreducible, that is, [Formula: see text] is stable. Under the assumption of explicit [Formula: see text]-conjecture, we further prove the stability of [Formula: see text] for the remaining values of [Formula: see text]. Also for [Formula: see text], if [Formula: see text] is reducible, then the number of irreducible factors of each iterate of [Formula: see text] is exactly [Formula: see text] for [Formula: see text].

On the number of irreducible factors with a given multiplicity in function fields

Let k≥1 be a natural number and f∈Fq[t] be a monic polynomial. Let ωk(f) denote the number of distinct monic irreducible factors of f with multiplicity k. We obtain asymptotic estimates for the first and the second moments of ωk(f) with k≥1. Moreover, we prove that the function ω1(f) has normal order log⁡(deg(f)) and also satisfies the Erdős-Kac Theorem. Finally, we prove that the functions ωk(f) with k≥2 do not have normal order.

Several classes of optimal p-ary cyclic codes with minimum distance four

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let p≥5 be an odd prime and m be a positive integer. Let C(1,e,s) denote the p-ary cyclic code with three nonzeros α, αe, and αs, where α is a generator of Fpm⁎, s=pm−12, and 2≤e≤pm−2. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal p-ary cyclic codes C(1,e,s) with parameters [pm−1,pm−2m−2,4]. Some known results on optimal quinary cyclic codes with parameters [5m−1,5m−2m−2,4] are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over F5m, we present two classes of optimal quinary cyclic codes C(1,e,s).

Parallel Codazzi tensors with submanifold applications

AbstractA decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.

Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank

We develop a new structural result for cohomogeneity one actions on (not necessarily irreducible) symmetric spaces of noncompact type and arbitrary rank. We apply this result to classify cohomogeneity one actions on the spaces SLn(R)/SOn, n≥2, up to orbit equivalence. We also reduce the classification problem on a reducible space to the classification on each one of its irreducible factors, which in particular allows to classify cohomogeneity one actions on any finite product of hyperbolic spaces.

On group actions on Riemann-Roch spaces of curves

Let G be a group acting on a compact Riemann surface X and D be a G-invariant divisor on X. The action of G on X induces a linear representation LG(D) of G on the Riemann-Roch space associated to D.In this paper we give some results on the decomposition of LG(D) as sum of complex irreducible representations of G, for D an effective non-special G-invariant divisor. In particular, we give explicit formulae for the multiplicity of each complex irreducible factor in LG(D). We work out some examples on well known families of curves.

Length-factoriality and pure irreducibility

An atomic monoid M is called length-factorial if for every non-invertible element , no two distinct factorizations of x into irreducibles have the same length (i.e., number of irreducible factors, counting repetitions). The notion of length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under the term “other-half-factoriality”: they used length-factoriality to provide a characterization of unique factorization domains. In this paper, we study length-factoriality in the more general context of commutative, cancellative monoids. In addition, we study factorization properties related to length-factoriality, namely, the PLS property (recently introduced by Chapman et al.) and bi-length-factoriality in the context of semirings.

Totally geodesic submanifolds in exceptional symmetric spaces

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.

Construction of quasi-cyclic self-dual codes over finite fields

ABSTRACT Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field F q of length mℓ for every positive even integer ℓ. In this paper, we study the case where x m − 1 has an arbitrary number of irreducible factors in F q [ x ] ; in the previous studies, only some special cases where x m − 1 has exactly two or three irreducible factors in F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over F q of length mℓ for any even positive integer ℓ, where we require that q ≡ 1 ( mod 4 ) if the index ℓ ≥ 6 . By implementation of our method, we obtain a new optimal binary self-dual code [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4.

Factoring Gleason polynomials modulo 2

Among the connected components of the interior of the Mandelbrot set are those that are hyperbolic. These components consist of parameters c∈ℂ for which the critical point z 0 =0 of f c :z↦z 2 +c is attracted to an attracting periodic cycle. Every hyperbolic component contains a unique center; that is, a parameter c for which the critical point z 0 is periodic. For a given n≥1, the Gleason polynomial for period n is the monic polynomial G n ∈ℤ[c] whose roots are exactly the centers of the hyperbolic components of period n. It is unknown if G n factors over ℤ. In this article, we factor G n modulo 2. We prove the following remarkable fact: the number of irreducible factors of G n modulo 2 is equal to the number of real roots of G n .

On the distribution of polynomials having a given number of irreducible factors over finite fields

Let $$q\geqslant 2$$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree n and have exactly k irreducible factors over the finite field $${\mathbb {F}}_q$$ . We also compare our results with the analogous existing ones in the integer case, where one studies all the natural numbers up to x with exactly k prime factors. In particular, we show that the number of monic polynomials grows at a surprisingly higher rate when k is a little larger than $$\log n$$ than what one would speculate from looking at the integer case.

An explicit formula for the A-polynomial of the knot with Conway's notation C(2n,4)

An explicit formula for the A-polynomial of the knot having Conway's notation C(2n,4) is computed up to repeated factors. Our polynomial contains exactly the same irreducible factors as the A-polynomial defined in [1].