975 research outputs found
The Cayley-Menger determinant is irreducible for
We prove that the Cayley-Menger determinant of an -dimensional simplex is
an absolutely irreducible polynomial for We also study the
irreducibility of polynomials associated to related geometric constructions.Comment: 7 pages, 4 figure
Quantitative equidistribution for the solutions of systems of sparse polynomial equations
For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},...,
x_n^{\pm1}] whose coefficients are not too big with respect to its directional
resultants, we show that the solutions in the algebraic n-th dimensional
complex torus of the system of equations f_1=\dots=f_n=0, are approximately
equidistributed near the unit polycircle. This generalizes to the multivariate
case a classical result due to Erdos and Turan on the distribution of the
arguments of the roots of a univariate polynomial. We apply this result to
bound the number of real roots of a system of Laurent polynomials, and to study
the asymptotic distribution of the roots of systems of Laurent polynomials with
integer coefficients, and of random systems of Laurent polynomials with complex
coefficients.Comment: 29 pages, 2 figures. Revised version, accepted for publication in the
American Journal of Mathematic
Factoring bivariate sparse (lacunary) polynomials
We present a deterministic algorithm for computing all irreducible factors of
degree of a given bivariate polynomial over an algebraic
number field and their multiplicities, whose running time is polynomial in
the bit length of the sparse encoding of the input and in . Moreover, we
show that the factors over \Qbarra of degree which are not binomials
can also be computed in time polynomial in the sparse length of the input and
in .Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a
multivariate version of Theorem 1 had independently been achieved by Erich
Kaltofen and Pascal Koira
Intrinsic palindromic numbers
We introduce a notion of palindromicity of a natural number which is
independent of the base. We study the existence and density of palindromic and
multiple palindromic numbers, and we raise several related questions.Comment: 6 pages, Latex2
- …