Géométrie des polynômes. Coût global moyen de la méthode de Newton
We consider a certain class of polynomials whose zeros are, all with one exception, close to the closed unit disk. We demonstrate that the Mahler measure can be employed to prove irreducibility of these polynomials over ℚ.
Let be a polynomial of degree at most which does not vanish in the disk , then for and , Boas and Rahman proved In this paper, we improve the above inequality for by involving some of the coefficients of the polynomial . Analogous result for the class of polynomials having no zero in is also given.
Kronecker sums and matricial norms are used in order to give a method for determining upper bounds for where is a latent root of a lambda-matrix. In particular, upper bounds for are obtained where is a zero of a polynomial with complex coefficients. The result is compared with other known bounds for .
MSC 2010: 30A10, 30C10, 30C80, 30D15, 41A17.In the present article, I point out serious errors in a paper published in Mathematica Balkanica three years ago. These errors seem to go unnoticed because some mathematicians are applying the results stated in this paper to prove other results, which should not continue.
Mathematics Subject Classification: 26A33, 47A60, 30C15.In this paper we treat the question of existence and uniqueness of solutions of linear fractional partial differential equations. Along examples we show that, due to the global definition of fractional derivatives, uniqueness is only sure in case of global initial conditions.