Factoring polynomials modulo special primes

We consider the problem of factoring polynomials overGF(p) for those prime numbersp for which all prime factors ofp– 1 are small. We show that if we have a primitivet-th root of unity for every primet dividingp– 1 then factoring polynomials overGF(p) can be done in deterministic polynomial time.Research partially supported by Hungarian National Foundation for Scientific Research, Grant 1812.     

A bivariate interval censorship model for partnership formation

We consider a statistical problem of estimating a bivariate age distribution of newly formed partnership. The study is motivated by a type of data that consist of uncensored, right-censored, left-censored, interval-censored and missing observations in the coordinates of a bivariate random vector. A model is proposed for formulating such type of data. A feasible algorithm to estimate the generalized MLE (GMLE) of the bivariate distribution function is also proposed. We establish asymptotic properties for the GMLE under a discrete assumption on the underlying distributions and apply the method to the data set.     

Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψ_υ:E(K_υ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψ_υ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψ_υ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψ_υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ_υ for complex archimedean υ.     

On zero-divisors of near-rings of polynomials

In this paper, we are interested to study zero-divisor properties of a 0-symmetric nearring of polynomials R₀[x], when R is a commutative ring. We show that for a reduced ring R, the set of all zero-divisors of R₀[x], namely Z(R₀[x]), is an ideal of R₀[x] if and only if Z(R) is an ideal of R and R has Property (A). For a non-reduced ring R, it is shown that Z(R₀[x]) is an ideal of Z(R₀[x]) if and only if ann_R({a, b}) ∩ N i?(R) ≠ 0, for each a, b ∈ Z(R). We also investigate the interplay between the algebraic properties of a 0-symmetric nearring of polynomials R₀[x] and the graph-theoretic properties of its zero-divisor graph. The undirected zero-divisor graph of R₀[x] is the graph Γ(R₀[x]) such that the vertices of Γ(R₀[x]) are all the non-zero zero-divisors of R₀[x] and two distinct vertices f and g are connected by an edge if and only if f ? g = 0 or g ? f = 0. Among other results, we give a complete characterization of the possible diameters of Γ(R₀[x]) in terms of the ideals of R. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its “multiplication” operation.     

The difference between the ordinary height and the canonical height on elliptic curves

We estimate the bounds for the difference between the ordinary height and the canonical height on elliptic curves over number fields. Our result is an improvement of the recent result of Cremona, Prickett, and Siksek [J.E. Cremona, M. Prickett, S. Siksek, Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006) 42-68]. Our bounds are usually sharper than the other known bounds.     

Lifting and recombination techniques for absolute factorization

In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.     

An explicit result of the sum of seven cubes

We prove that every integer ≥ exp(524) is a sum of seven non negative cubes.     

On Eisenstein polynomials and zeta polynomials

Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.     

Galois theory, discriminants and torsion subgroup of elliptic curves

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.     

Bounds for resultants of univariate and bivariate polynomials

The paper considers bounds on the size of the resultant for univariate and bivariate polynomials. For univariate polynomials we also extend the traditional representation of the resultant by the zeros of the argument polynomials to formal resultants, defined as the determinants of the Sylvester matrix for a pair of polynomials whose actual degree may be lower than their formal degree due to vanishing leading coefficients. For bivariate polynomials, the resultant is a univariate polynomial resulting by the elimination of one of the variables, and our main result is a bound on the largest coefficient of this univariate polynomial. We bring a simple example that shows that our bound is attainable and that a previous sharper bound is not correct.     

On Certain Subcovers of the Hermitian Curve

We present a simple construction that gives explicit equations for certain subcovers of the Hermitian curve. We show that maximal curves with a certain type of defining equations are covered by the Hermitian curve.     

Explicit determination of generalized symmetric and alternating Galois groups

This paper shows a probabilistic algorithm to decide whether the Galois group of a given irreducible polynomial with rational coefficients is the generalized symmetric group C_p?S_m or the generalized alternating group C_p?A_m. In the affirmative case, we give generators of the group with their action on the set of roots of the polynomial.     

Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group $S_n$ on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on ${\mathbb{R}}^{2n}/S_n$ more efficiently than known polynomials. Moreover, we present special cases for $r\ge 3$ where the basic polynomials separate orbits.     

Fast conversion algorithms for orthogonal polynomials

We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation.     

Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p?7 which is unramified in K. Given an elliptic curve A/Q with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert-Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Irreducible multivariate polynomials obtained from polynomials in fewer variables

We provide several methods to construct irreducible multivariate polynomials from irreducible polynomials in a smaller number of variables.     

On elliptic units and p-adic Galois representations attached to elliptic curves

Let K be a quadratic imaginary number field with discriminant D_K≠-3,-4 and class number one. Fix a prime p?7 which is not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.