Properties that characterize Gaussian periods and cyclotomic numbers

Let be a prime number, a -th primitive root of 1 and the periods of degree of . Write with . Several characterizations of the numbers and (or, equivalently, of the cyclotomic numbers of order ) are given in terms of systems of equations they satisfy and a condition on the linear independence, over , of the or on the irreducibility, over , of the characteristic polynomial of the matrix .

     

Jacobi sums and new families of irreducible polynomials of Gaussian periods

Let 2$">, an -th primitive root of 1, mod a prime number, a primitive root modulo and . We study the Jacobi sums , , where is the least nonnegative integer such that mod . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families , , of irreducible polynomials of Gaussian periods, , of degree , where is a suitable set of primes mod . We exhibit examples of such families for several small values of , and give a MAPLE program to construct more of them.

     

Period polynomials for generalized cyclotomic periods

The theory of cyclotomic period polynomials is developed for general periods of an arbitrary modulus, extending known results for the Gauss periods of prime modulus. Primes dividing the discriminant of the period polynomial are investigated, as are those primes dividing values of the period polynomial.Author has NSF grant MCS-8101860     

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.     

Chebyshev polynomials,Catalan numbers,and tridiagonal matrices

Theoretical and Mathematical Physics - We establish a relation between linear second-order difference equations corresponding to Chebyshev polynomials and Catalan numbers. The latter are the limit...     

Coefficients of ternary cyclotomic polynomials

It is customary to define a cyclotomic polynomial Φn(x) to be ternary if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter (1968, 1971) [3] and [4] conjectured that . In 2008, Yves Gallot and Pieter Moree (2009) [6] showed that the conjecture is false for every p≥11, and they proposed the Corrected Beiter conjecture: . Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p=7.     

On difference polynomials and hereditarily irreducible polynomials

A difference polynomial is one of the form P(x, y) = p(x) ? q(y). Another proof is given of the fact that every difference polynomial has a connected zero set, and this theorem is applied to give an irreducibility criterion for difference polynomials. Some earlier problems about hereditarily irreducible polynomials (HIPs) are solved. For example, P(x, y) is called a HIP (two-variable case) if P(a(x), b(y)) is always irreducible, and it is shown that such two-variable HIPs actually exist.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

Class numbers of cyclotomic fields

In the first part of the paper we show how to construct real cyclotomic fields with large class numbers. If the GRH holds then the class number h_p⁺ of the pth real cyclotomic field satisfies h_p⁺ > p for the prime p = 11290018777. If we allow n to be composite we have, unconditionally, that $\text{h}{}_{\mathrm{n}}{}^{\mathrm{+}}\text{> n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; \epsilon </mtext>}}$ for infinitely many n. In the second part of the paper we show that if l ?= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then $\text{l}\text{2}$ (l, if l is odd) divides the class number h_n⁺ of the nth cyclotomic field. If the primes are congruent to 1 mod 4l then 2l divides h_n⁺.     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

On testing the divisibility of lacunary polynomials by cyclotomic polynomials

An algorithm is described that determines whether a given polynomial with integer coefficients has a cyclotomic factor. The algorithm is intended to be used for sparse polynomials given as a sequence of coefficient-exponent pairs. A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.

     

Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations

A general method of constructing families of cyclic polynomials over with more than one parameter will be discussed, which may be called a geometric generalization of the Gaussian period relations. Using this, we obtain explicit multi-parametric families of cyclic polynomials over of degree . We also give a simple family of cyclic polynomials with one parameter in each case, by specializing our parameters.

     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.     

A survey on coefficients of cyclotomic polynomials

Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their coefficients have been intensively studied by several authors, and in the last 10 years there has been a burst of activity in this field of research. This concise survey attempts to collect the main results regarding the coefficients of the cyclotomic polynomials and to provide all the relevant references to their proofs.