Luiz Emilio Allem

Extracting sparse factors from multivariate integral polynomials

In this paper we present a new algorithm for extracting sparse factors from multivariate integral polynomials. The method hinges on a new type of substitution, which reduces multivariate integral polynomials to bivariate polynomials over finite fields and keeps the sparsity of the polynomial. We retrieve the multivariate sparse factors, term by term, using discrete logarithms. We show that our method is really effective when used for factoring multivariate polynomials that have only sparse factors and when used to extract sparse factors of multivariate polynomials that may also have dense factors.

Suporte de grafos unicíclicos

Neste trabalho obtemos uma caracterizacao dos grafos uniciclicos singulares atraves do suporte de suas ´arvores pendentes. Alem disso, mostramos que o suporte de algumas ´arvores pendentes de um grafo G uniciclico do tipo I esta contido no suporte de G.

Multiplicity of eigenvalues of cographs

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for the multiplicity.The energy of a graph is defined as the sum of absolute values of the eigenvalues. A graph G on n vertices is said to be borderenergetic if its energy equals the energy of the complete graph Kn. We present families of non-cospectral and borderenergetic cographs.

Functional Decomposition Using Principal Subfields

Let f ∈ K(t) be a univariate rational function. It is well known that any non-trivial decomposition g o h, with g,h ∈ K(t), corresponds to a non-trivial subfield K(f(t)) ⊊ L ⊊ K(t) and vice-versa. In this paper we use the idea of principal subfields and fast subfield-intersection techniques to compute the subfield lattice of K(t)/K(f(t)). This yields a Las Vegas type algorithm with improved complexity and better run times for finding all non-equivalent complete decompositions of f.

A pre-test for factoring bivariate polynomials with coefficients in F 2

Abstract We introduce a pre-test for bivariate polynomial factorization over F 2 , which combines the basic structure of an algorithm due to Lecerf (2010) [14] with ideas of Gao (2003) [5] .

Q-integralidade do Grafo Total de Kn

Grafos Q-integrais sao grafos cujo espectro em relacao a matriz laplaciana sem sinal e constituido somente por numeros inteiros. A caracterizacao geral dos grafos com esta propriedade e um problema bem dificil. Ao mesmo tempo que se busca grafos Q-integrais tambem e interessante encontrar operacoes que preservem a Q-integralidade dos grafos, ou seja, operacoes que quando aplicadas em grafos Q-integrais gerem novos grafos Q-integrais. Sabemos que os grafos completos sao Q-integrais [1]. Nossa contribuicao consiste em mostrar que a operacao grafo total quando aplicada em um grafo completo preserva a Q-integralidade do mesmo, ou seja, gera um novo grafo Q-integral.

Gcd of multivariate polynomials via Newton polytopes

We study geometric criteria to determine coprimality between multivariate polynomials. Our main contribution is the development of a polynomial-time algorithm (on the number of monomials) that detects coprimality of multivariate polynomials using Newton polytopes. We also show how to construct the gcd of two bivariate polynomials using their Newton polygons.