An 1861 theorem of Ch. Hermite [He] asserts that every field extension (and more generally, every étale algebra) E/F of degree 5 can be generated by an element a ∈ E whose minimal polynomial is of the form f(x) = x + b2x 3 + b4x+ b5 . Equivalently, trE/F (a) = trE/F (a ) = 0. A similar result for étale algebras of degree 6 was proved by P. Joubert in 1867; see [Jo]. It is natural to ask whether...

There exist various well-known characterizations of sets of numbers recognizable by a nite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties.

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I . Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I . It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a “resolved problem”. But being the key of...

Let L = K() be an abelian extension of degree n of a number eld K, given by the minimal polynomial of over K. We describe an algorithm for computing the local Artin map associated to the extension L=K at a nite or innnite prime v of K. We apply this algorithm to decide if a nonzero a 2 K is a norm from L, assuming that L=K is cyclic.

Generalized GIPSCAL, like DEDICOM, is a model for the analysis of square asymmetric tables. It is a special case of DEDICOM, but unlike DEDICOM, it ensures the nonnegative definiteness (nnd) of the model matrix, thereby allowing a spatial representation of the asymmetric relationships among “objects”. A fast convergent algorithm was developed for GIPSCAL with acceleration by the minimal polynom...

Let n1 and n2 be two distinct primes with gcd(n1 − 1, n2 − 1) = 4. In this paper, we compute the autocorrelation values of generalized cyclotomic sequence of order 4. Our results show that this sequence can have very good autocorrelation property. We determine the linear complexity and minimal polynomial of the generalized cyclotomic sequence over GF(q) where q = pm and p is an odd prime. Our r...

AGarsia number is an algebraic integer of norm ±2 such that all of the roots of its minimal polynomial are strictly greater than 1 in absolute value. Little is known about the structure of the set of Garsia numbers. The only known limit point of positive real Garsia numbers was 1 (given, for example, by the set of Garsia numbers 21/n). Despite this, there was no known interval of [1,2] where th...

Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we ...

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show...