Number Of Irreducible Factors Research Articles

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.

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.

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.

Number of irreducible factors and degree in divisor graph of Zp[x,n

The divisor graph, denoted by D(Z p [x,n]), is the graph whose vertex set is the set of all polynomials of degree at most n whose coefficients are from field Z p and its any two distinct vertices are adjacent if one is a divisor of the other. In this paper, (i) we determine the degree of each vertex of D(Z p [x,3]) and also discuss its girth, size, degree sequence, irregularity index etc. (ii) We also establish that two polynomials of same degree k in Z p [x,n] having different number of irreducible factors, the one with fewer number of irreducible factors has smaller degree. (iii) Further, if two polynomials of same degree k in Z p [x,n] having same number of irreducible factors but different number of distinct irreducible factors, the one with fewer number of distinct irreducible factors has smaller degree.

Number of irreducible factors and degree in divisor graph of Zp[x,n

The divisor graph, denoted by D(Z p [x,n]), is the graph whose vertex set is the set of all polynomials of degree at most n whose coefficients are from field Z p and its any two distinct vertices are adjacent if one is a divisor of the other. In this paper, (i) we determine the degree of each vertex of D(Z p [x,3]) and also discuss its girth, size, degree sequence, irregularity index etc. (ii) We also establish that two polynomials of same degree k in Z p [x,n] having different number of irreducible factors, the one with fewer number of irreducible factors has smaller degree. (iii) Further, if two polynomials of same degree k in Z p [x,n] having same number of irreducible factors but different number of distinct irreducible factors, the one with fewer number of distinct irreducible factors has smaller degree.

Orthogonal bi-invariant complex structures on metric Lie algebras

This paper studies how many orthogonal bi-invariant complex structures exist on a metric Lie algebra over the real numbers. Recently, it was shown that irreducible Lie algebras which are additionally 2-step nilpotent admit at most one orthogonal bi-invariant complex structure up to sign. The main result generalizes this statement to metric Lie algebras with any number of irreducible factors and which are not necessarily 2-step nilpotent. It states that there are either 0 or $$2^k$$ such complex structures, with k the number of irreducible factors of the metric Lie algebra. The motivation for this problem comes from differential geometry, for instance to construct non-parallel Killing-Yano 2-forms on nilmanifolds or to describe the compact Chern-flat quasi-Kahler manifolds. The main tool we develop is the unique orthogonal decomposition into irreducible factors for metric Lie algebras with no non-trivial abelian factor. This is a generalization of a recent result which only deals with nilpotent Lie algebras over the real numbers. Not only do we apply this fact to describe the orthogonal bi-invariant complex structures on a given metric Lie algebra, but it also gives us a method to study different inner products on a given Lie algebra, computing the number of irreducible factors and orthogonal bi-invariant complex structures for varying inner products.

Symbol Separation in Double Occurrence Words

A double occurrence word (DOW) is a word in which every symbol appears exactly twice. We define the symbol separation of a DOW [Formula: see text] to be the number of letters between the two copies of a symbol, and the separation of [Formula: see text] to be the sum of separations over all symbols in [Formula: see text]. We then analyze relationship among size, reducibility and separation of DOWs. Specifically, we provide tight bounds of separations of DOWs with a given size and characterize the words that attain those bounds. We show that all separation numbers within the bounds can be realized. We present recursive formulas for counting the numbers of DOWs with a given separation under various restrictions, such as the number of irreducible factors. These formulas can be obtained by inductive construction of all DOWs with the given separation.

On the factorization of iterated polynomials

Let $\mathbb{F}\_q$ be the finite field with $q$ elements and $f, g\in \mathbb{F}\_q\[x]$ be polynomials of degree at least one. This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the iterated polynomials $f(g^{(n)}(x))$ over $\mathbb{F}\_q$, such as the largest degree of an irreducible factor and the number of irreducible factors. In particular, we provide significant improvements on the results of D. Gómez-Pérez, A. Ostafe and I. Shparlinski (2014).

On Mersenne polynomials over [formula omitted

We work over the field with two elements. We establish a new correspondence between Mersenne polynomials and trinomials so that corresponding polynomials have the same number of irreducible factors. This allows us to get a partial but nontrivial result about the factorization of M2h+1+1, for a Mersenne prime M and for a positive integer h.

Optimized AKS Primality Testing: A Fluctuation Theory Perspective

The AKS algorithm is an important breakthrough in showing that primality testing of an integer can be done in polynomial time. In this paper, we study the optimization of its runtime. Namely, given a finite cardinality set of alphabets of a deterministic polynomial runtime Turing machine and the number of strings of an arbitrary input integer whose primality is to be tested as the system parameters, we consider the randomized AKS primality testing function as the objective function. Under randomization of the system parameters, we have shown that there are definite signatures of the local and global instabilities in the AKS algorithm. We observe that instabilities occur at the extreme limits of the parameters. It is worth mentioning that Fermat’s little theorem and Chinese remaindering help with the determination of the underlying stability domains. On the other hand, in the realm of the randomization theory, our study offers fluctuation theory structures of the AKS primality testing of an integer through its maximum number of irreducible factors. Finally, our optimization theory analysis anticipates a class of real-world applications for future research and developments, including optimal online security, system optimization and its performance improvements, (de)randomization techniques, and beyond, e.g., polynomial time primality testing, identity testing, machine learning, scientific computing, coding theory, and other stimulating optimization problems in a random environment.

On the local k-elasticities of Puiseux monoids

If [Formula: see text] is an atomic monoid and [Formula: see text] is a nonzero non-unit element of [Formula: see text], then the set of lengths [Formula: see text] of [Formula: see text] is the set of all possible lengths of factorizations of [Formula: see text], where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O’Neil studied the sets of elasticities [Formula: see text] of Puiseux monoids [Formula: see text]. Here, we take this study a step further and explore the local [Formula: see text]-elasticities of the same class of monoids. We find conditions under which Puiseux monoids have all their local elasticities finite as well as conditions under which they have infinite local [Formula: see text]-elasticities for sufficiently large [Formula: see text]. Finally, we focus our study of the [Formula: see text]-elasticities on the class of primary Puiseux monoids, proving that they have finite local [Formula: see text]-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain [Formula: see text] as a limit point.

The arithmetic of consecutive polynomial sequences over finite fields

Motivated by a question of van der Poorten about the existence of an infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of irreducible factors, the number of irreducible factors, as well as for the number of such sequences of fixed length in which all the polynomials are irreducible.

The Reducibility of a Special Binary Pentanomial

Swan’s theorem determines the parity of the number of irreducible factors of a binary trinomial. In this work, we study the parity of the number of irreducible factors for a special binary pentanomial with even degree xm+xn1+xn2+xn3+1, where 0<n3<n2<n1≤m/2, and exactly one of n1,n2, and n3 is odd. This kind of irreducible pentanomials can be used for a fast implementation of trace and square root computations in finite fields of characteristic 2.

ON IRREDUCIBLE FACTORS OF THE POLYNOMIAL f(x) - g(y)

In this paper, all irreducible factors of bivariate polynomials of the form f(x) - g(y) over an arbitrary field are described. It also proves that the number of irreducible factors of f(x) - g(y) (counting multiplicities) does not exceed the greatest common divisor of the degrees of f(x) and g(y), yielding a well known result of Tverberg regarding the irreducibility of f(x) - g(y). It proves that if f(x) and g(y) are non-constant polynomials with coefficients in the field ℚ of rational numbers and deg f(x) is a prime number, then f(x) - g(y) is a product of at most two irreducible polynomials over ℚ. This contributes to a problem raised by Cassels which asks for the polynomials f, such that the polynomial [Formula: see text] is reducible.

A note on the reducibility of binary affine polynomials

Stickelberger---Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over $${\mathbb{F}_2}$$ with degree >1, where A(x) = L(x) + 1 and $${L(x)=\sum_{i=0}^{n}{x^{2^i}}}$$ is a linearized polynomial over $${\mathbb{F}_2}$$ , is reducible except x 2 + x + 1 and x 4 + x + 1. We also give some explicit factors of some special affine pentanomials over $${\mathbb{F}_2}$$ .

Power series extensions of half-factorial domains

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R , and independent variables, Y and T , R [ [ Y ] ] [ [ T ] ] is not HFD even when R [ [ Y ] ] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.

Irreducibility results for compositions of polynomials in several variables

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.

Subspaces and polynomial factorizations over finite fields

Recently Niederreiter described a new method for factoring polynomials over finite fields. As with the Berlekamp technique, the method requires the construction of a linear subspace whose dimension is precisely the number of irreducible factors of the polynomial being considered. This paper explores the connection between these subspaces and gives a characterization of other subspaces having properties which are similar.

Random Matrices and Brownian Motion

For T ∈ GLn (Fq), let Ωn (t, T) be the number of irreducible factors of degree less than or equal to nt in the characteristic polynomial of T. Letand suppose T is chosen from G Ln(Fq) at random uniformly. We prove that the stochastic process ≺Zn(t)≻t∈[0, 1] converges to the standard Brownian motion process W(t), as n → ∞.

Factoring Rational Polynomials over the Complex Numbers

NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables (d, c, and n, respectively). If n is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor. After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of $(P(z_1 ,z_2 ) = 0)$ correspond to the connected components of the real surface (or complex curve) $P(z_1 ,z_2 ) = 0$ minus its singular points. In finding the number of connected components of the surface $P = 0$, the surface is projected to the the $z_2 $-plane. The singular points of $P(z_1 ,z_2 )$ lie over the projection’s critical values. The inverse image of a grid isolating the critical values in the $z_2 $-plane lifts to a one-dimensional real curve skeleton on the surface $(P = 0)$ whose number of connected components is precisely the number of connected components of $P = 0$ minus its singular points. The connectivity of this curve skeleton is constructed symbolically using Sturm sequences associated with the various polynomials defining these maps. Given the number of irreducible factors and their degrees, the actual factors can be reconstructed using the recent result of Neff [Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 152–162] on finding zeros of one-variable polynomials in NC.